On the Equivalence Functional Relation Estimated for Vector-Valued Singular Integrals with Semigroups of Operators

Republic of the Sudan Ministry of High Education and Scientific Research University of Nile Valley Faculty of Postgraduate Studies

On The Equivalence Functional Relation Estimated For Vector-Valued Singular Integrals with Semigroups of Operators.

A Thesis Submitted in Partial Fulfillment for the Degree of M.sc in Mathematics

Prepared by: Mohammed Salih Mukhtar Dahab Ahmed

Supervised by: Dr. Adam Abedalla Abbakar Hassan

December, 2009

اآلية

ق ال اهلل تعالي ِ ِ }يَْومَ نَطْوِي السََّماء َكطَ ِّي السِّجلِّ لْل ُكتُ ِب كََما بَدَأْنَ ا أَوَّلَ ِ ِِ خَْل ٍق نُّعيدُهُ َو ْعداً عَل َْينَ ا إِن َّا ُكن َّا ف َاعلي َن {األنبياء401

وق ال اهلل تعالي أيضا ِ ِ }َوَما قَدَُروا اللَّوَ َح َّق قَ ْدرِه َواْألَْر ُض َجميعا ً قَْب َضتُوُ يَْومَ ِ ِ ِ ِ ِِ اْلقيَ اَمة َوالسَّماَوا ُت َمطْوِي َّا ٌت بيَمينو ُسْبَحانَوُ َوتََعالَى عَمَّا يُ ْشرِ ُكو َن {الزمر76

وق ال اهلل تعالي أيضا ِ ٍ ِ }َوالسََّماء بَنَْينَ ا َىا بأَْيد َوإِن َّا لَُموسُعو َن * َواْألَْر َض فََر ْشنَ ا َىا فَنِعم اْلما ِىدون {الذاريات16، 14 ْ َ َ ُ َ صدق اهلل العظيم

Dedication TO My parents who survived for me to be

My great thanks after my God to my supervisor Dr. Adam Abedd-

Alla Abbakar who suggested this title , for his wise advice and guidance. Thanks are Also due to the library staff of the Facluty of

Mathematical Sciences U.of.K and especially for Mr. Mohammed

Yasein the secretary. Also are extended to the teaching staff and administrators of the M.sc program at the Nile Valley University

Omdurman branch.

ii Abstract

In this thesis we have present a mathematical techniques in estimation of vector – valued singular integral by using equivalence functional relations with semigroups of operators.

This thesis contains four chapters. The first chapter is the function and the functional relations. The second chapter with the title vector – valued singular integrals. Chapter three under the title of semigroups and semigroups of operators. The last , chapter four, in which the estimation of the singular integragrls have been done by developing a generalized Littlewood-paley theory for semigroups acting on Lp -spaces of functions with values in uniformly convex or smooth Banach spaces, this general theory becomes more satisfactory (and easier to be handled) in virtue of the theory of vector-valued Calder´on-Zygmund singular integral operators.

iii

الخالصة

قددت في ددًيهدد ي لبحددبيب ددتي لات ٌدديفي ل ٌيمددٌدي ل ددا ت دي ددًياتددتٌ ي لامددي في

ل ف تةيأوي لشي ةيي في لتٌ ددي تاايهٌددي لادًيا دا تعي ل قديفي لت لٌددي د ي دثر في

شبهي لز .يٌحاوييهد ي لبحدبيليدًيأ ب ددي دو ي,ي لف د ي وو يب دو اي لت لدديوي

ل قيفي لت لٌدي,يو لف ي لري ًيٌ ا تي لفمديا في ل ااهدديو لامدي في ل فد تةي

أوي لشددي ةيي يآفي لتٌ دددي تاايهٌدددي.ي لف دد ي لريلددبيلدداي ددثر فيشددبهي لز دد ي,يأ دديي

لبيبي ل ب يو و ٌ ياعي ٌهياتتٌ يايكي لامدي فيلدايي ٌدطيايدوٌ ي ظ ٌدديلٌايدووتي

–يبٌيى(Littlewood-Paley theory) يلشبهي لز ي لاًيادثر ي دًيتو ي مديا في

لبٌ ددٌج)Lebeseque space)ي في لتٌ دددي ددًي بدديفيبي ددي (Banach space)ي

له ت ٌدي ل اظ دي ل حتبد(convex)يأويي لافيميٌدي ل يل د)smooth(يلا دبحيهد هي

ل ظ ٌدي في فسي ٌزةي ثر فيميلٌت واي–يز ٌت و تييليامي في ل ف تةيأوي لشدي ةيي

في لتٌ دددي تاايهٌدددي Calderon-Zagmound vector-valued singular)

(integrals ي

v List of contents

Dedication…………………………………………………………i Acknoweledgement…………………………………………….…ii Abstract…………………………………………………………...iii Abstract (Arabic)………………………………………………....iv List of contents……..……………………………….…………….v List of symbols and abbreviations……………………………….vii Introduction……………………………………………….………x

Chapter one

1-Function and functional relations.….…………………….……..1 11  The function…………………………………………………………..1 12  The functional relations…………………………………………6 13  The equivalence functional relations……………………..18

Chapter two 2-Vector –valued singular integrals……………………………..27 21 Vector space……………………………………………....28 22 Topological vector space…………………………………29 23 Kernel mathematic………………………………………..32 24 Convolution………………………………………………………..40 25 Integral transform……………………………………………….42 26 Singular integrals………………………………………………..43 27 Hardy – Littlewood maximal function…………………..47 28 Littlewood-Paley theory……………………………………….50 29 Fefferman - Stein function and vector-valued operators...53 v Chapter three

3-Semigroups and semigroups of operators……………………55 31 The semigroups…………………………………………...56 32 Semigroups action………………………………….…….60 33 Transformation semigroup………………………………..63 34 Inverse semigroup………………………………………...64 35 Semigroups of operators………………………………….67 36  Hille–Yosida theorem……………………………………69

37  C0-semigroup……………………………………………72 38 Special integral operators…………………………………77

Chapter four

4-Estimation of vector-valued singular integrals with semigroups of operators………………………………………………...……87 41 Developing a generalized Littlewood-Paley theory for semigroups acting on Lp -spaces…………………………………89 42 One-sided vector-valued Littlewood-Paley-Stein inequalities for semigroups………………………………………94 43 Duality…………………………………………………..110 44  Poisson semigroup on Rn ……………………………..113 45  Continuation of Poisson semigroup on Rn ……………..121 46 Ornstein-Uhlenbeck semigroup…………………………126 47 Almost sure finiteness…………………………………..130 References…………………………….………………………..137

vi List of symbols and abbreviations

f Function

CC, 0   Set of continuous functions

R \0  Domain of real numbers except zero

 X  Equivalence class

: Binary relation

,,,,,,:;     Equivalence relations

ConX Complete lattice

ker Mathematical kernel

Rn N-dimensional Euclidean space

(CZO) Calderón-Zygmund Operator

(UMD) unconditional martingale differences

 The norm

Lp  Lebesgue spaces

W kp, Sobolev spaces

vii P Poisson kernel

H p Hardy space

M Maximal function

gf  Littlewood-Paley g- function

f # The sharp maximal function

Q Cubes

H Hilbert space

B Banach space

Ttt0 Diffusion semigroup

W Weierstrass Operators

 Gradient operator

 The Laplace operator

 n Gaussian measure

L Ornstein–Uhlenbeck operator

BMO Bounded mean oscillator

G The classical Littlewood-Paley g -function

viii

Pt t0 The subordinated Poisson semigroup

A The infinitesimal generator of semigroup

q g -function” associated to subordinated Poisson semigroup

T Torus

Gq n -dimensional generalized g-function

n Ott0 The Ornstein-Uhlenbeck semigroup on R

O  t t0 Poisson semigroup subordinated to the Ornstein-Uhlenbeck semigroup on

gq Littlewood-Paley g -function associated to

En Monotone sequence of conditional Expectations

n Pt Poisson kernels both on and on R dμ Lebesgue measure

Pr Poisson kernels on

Introduction

In this thesis we have to present a mathematical technique in estimation of vector – valued singular integral by using equivalence functional relations with semigroups of operators. This thesis contains four chapters. the first chapter is the function and the functional relations, in which we have introduced basic results of the function its (definition, representation, notation, properties and continuity), the functional relations (injection, bijection, surjection, isometry and homomorphism) and the equivalence functional relations with the known notations ,,,,,    that used as tools to compare between mathematical terms or structures and help us to estimate. In the second chapter with the title vector – valued singular integrals we start with the vector space and topological vector spaces and their properties examples are Banach spaces, Hilbert spaces and Sobolev spaces. Those spaces are underlying valued spaces for the functions define and bounded on them. After this we view the technique to achieve the singular integral by giving definition , meaning ,and properties of the kernels like (Calderon – Zagmund kernels , Poisson kernels), then we introduce the convolution , where the singular integral is an integral transform given by convolution of the function against the kernels , also the integrable function ( Lebesque measurable functions and Bochner integrals) , integral transform , Hilbert transform in harmonic analysis in the L2 space, from this view we have studied the Calderon – Zagmund singular operator with kernels in Rn . The rest of this chapter studies the development of the function to maximal function, Hrdy-Littlewood maximal functions, Littlewood – Paley theory and Fefferman Stein extension to the latter theory. That sharp maximal function f # is defined as regarding singular integrals. x In chapter three under the title of semigroups of operators, we exhibits the semigroups with it’s definition, types, structure, action, and it’s properties. Then we introduce the semigroup operators its definition and generation, under the fundamental Hille–Yosida theorem we study the C0-semigroup and its infinitesimal operator, also known as a (strongly continuous) one-parameter semigroup, the transformation and the analytical semigroup are special type of the C0-semigroup Then we define the symmetric diffusion semigroup ,it is a collection of linear operators Ttt0 defined on Ldp ,   over a measure space ,d  which is used in the solution of the partial differential equation and share the same application with the singular integral. At the end of this chapter we introduce some integral operators with semigroups properties, The Weierstrass operators, Poisson operator, and the Ornstein– Uhlenbeck operator used in the development of Littlewood-Paley generalized function for the semigroups in Rn spaces.

The last , chapter four, in which the estimation of the singular integragrls have been done by developing a generalized Littlewood-paley theory for semigroups acting on Lp -spaces of functions. It is well known that martingale inequalities involving square function are closely related to the corresponding inequalities concerning the Littlewood-Paley or Lusin square function in harmonic analysis. It is in this spirit that a generalized 49 Littlewood-Paley theory is developed in [XU] . The main goal is to extend the results in [XU] to general symmetric diffusion semigroups, and thus to develop a generalized Littlewood-Paley theory for these semigroups on Lp -spaces of functions. We characterize, in the vector-valued setting, the validity of the one-

xi sided inequalities concerning the generalized Littlewood-Paley- Stein g -function associated with a subordinated Poisson symmetric diffusion semigroup by the martingale co-type and type properties of the underlying Banach space. We showed in section44  and 45 in the case for the usual Poisson semigroup and the Poisson semigroup subordinated to the Ornstein-Uhlenbeck semigroup on Rn , this general theory becomes more satisfactory (and easier to be handled) in virtue of the theory of vector-valued Calder´on-Zygmund singular integral operators.

xii

CHAPTER One

CHAPTER ONE

FUNCTION AND FUNCTIONAL RELATIONS

In This chapter we introduce preliminaries for this thesis which gives essential information about the function (definition, representation, notation, properties and continuity),the functional relations (injection, bijection, surjection, isometry and homomorphism) and the equivalence functional relation with the notation,,,,,:;    

11 The function (mapping): 1 1.1.1Definition: (A function is a rule that produces correspondences between two sets of elements say  AB, such that to each element of the first set ()A there exist one and only one element of the second setB

1 X

2 Y

3 Z

The elements of the first set called the domain and the elements of the second set called the codomain.

Ways of representing function: If f is the name of the function defined by the equation yx  then instead of the formal representation we write: f: y 2 x 1 (Rule of correspondence)

Or f: x , y y 2 x 1 (Set of order pairs). Or simply f x  2 x 1 , or f: x y (function notation).

1.1.2Definition: 1 Let  XY, be two sets for each element x of X there exist only one element y of Y (unless otherwise stated however), it will assume here that for each element x in the domain of , f (x) contains one element y such a function is said to be single valued.

1.1.3Definition: The unique function over a set that maps each element to itself is called the identity function for , and typically denoted by idX Each set has it is own identity function, so subscript cannot be omitted unless the set can be inferred from the text. Under composition, an identity function is neutral: if f is any function from toY , then

f idX  f And idX f f

1.1.4Definition: If f is function from set to set Y then an inverse function for f, denoted by f -1, is the opposite direction, from Y to , not every function has an inverse; those that do are called invertible.

Pointwise operations:

If f: X Rand g: X R are functions with a common domain and a common codomain a ring R, then one can define a sum function  f g: X R and the product function f g: X R as:  f g x  f x  g x ,  f g x  f x  g x

1.1.5Definition: 1

Let XYZ,, be set such that the function f: X Y and the function g: Y Z , then the composite function define as f g: X Z

1.1.6Definition: Let XY, be subsets of R, then for some xXε and some yYε with scalars αβ, then the function f is said to be linear if: fα x β y  α f x  β f y 1

1.1.7Definition: 2 A function is said to be continuous at a point c of its domain if when the following two requirements are satisfied: i. fc  Is defined. ii. The limit of fx as x approach cmust be exist and equal to . iii. Everywhere continuous function: We say that a function is continuous everywhere or simply continuous if it is continuous at every point of its domain.

The notationC: The notation or C0 is sometimes used to denote the set of all continuous functions with a domain .

Cauchy (epsilon – delta) definition of continuity 2

1.1.8Definition: The function f is said to be continuous at point c if and only if the following holds:

For any number  >0 however small there exist some number δ >0 such that for all x in the domain c δ < x

Heine definition of continuity 2

1.1.9Definition: The following definition of continuity due to Heine a real function f is continuous if for any sequence xn such that: lim xx  , it holds that , lim f x  f x n n 0 n n  0  We assume that all points xxn, 0 belong to the domain of fx .

Directional continuity

1.1.10Definition: A function may happen to be continuous in only one direction, either from the right or from the left a right continuous function is the function, which continues at all, points when approached from the right. A right formal definition: a function is said to be right continuous at point c if and only if the following holds: For any number >0 however small there exist some number >0 such that for all x in the domain < < the values of satisfies < < Likewise a left continuous function is the function, which is continuous at all, points when approached from the left. A function is continuous if and only if it is both right and left continuous

Continuous function between topological spaces 2

1.1.11Definition:

If X and Y are two topological spaces and f is a function such that f: X Y is continuous if and only if for every open set VY the inverse image f1  V   xεε X f() x V is open.

Continuous function neighborhood

1.1.12Definition:

Suppose we have a function f: X Y , where X and Y are topological spaces . We say f is continuous at x for some xX if for any neighborhood V of f(x), there is a neighborhood U of x such that f() U V .

Facts about continuity: If two functions f and g are continuous then fg and fg are continuous and if g  0 for all x in the domain, then fgis also continuous the composite function of two continuous functions fgis continuous.

Examples: 2

i. The rational functions, the exponential function, the logarithms functions, square root functions, absolute value functions, are all continuous functions. ii. Consider the function f( x ) 1  x and the function g( x ) sin( x ) ( x ) neither function is defined at x  0 and each has a domain R 0 of real number except 0, so equation of continuity at does not arise, since it is not in the domain, on the other hand the limit at is 1, g can extended continuity to Rby defining its limit at 0 to be 1 2

12 The functional relations:

The Injective function: 3

1.2.1Definition: In mathematics, an injective function is a function, which associates distinct arguments to distinct values. More precisely a function f is said to be injective if it maps distinct x in the domain to distinct y in the codomain, such f() x y. But another way is injective if f()() a f b implies ab or ()ab implies that f()() a f b , for any ab, in the domain. The diagram below shows the injective functional relation:

 X  f ( a ) Y  a f (b)

f(c) b

C f(d)

An injective function is called an injection, and also said to be information – preserving or one -to- one function (however, the latter name is best avoided, since sometimes authors understand it to mean a one to one correspondence, i.e., a function). A function f that not injective is some times called many to one, however, this name too is best avoided , since it is some times used to mean, “ single valued “, and i. e. each argument is mapped to at most one value. Examples and counter examples: 3

i. For any set, the identity function on X is injective. ii. The function f: R R defined by g() x x2 is not injective because (for example)gg  , however, if g is redefined so that its domain is non-negative real numbers0,, then g is injective. iii. The exponential function exp :R R : x ex is injective. iv. The natural logarithm function Ln: 0,  R : x  Ln ( x )is injective.

Injection can be undone:

Functions with left inverses (often called section) are always injection. That is to say, f: X Y .If there exists a function g: Y X such that, for every g f x  x ( can be undone by ) then is injective. Conversely, it is usually assume that every injection with non – empty domain has a left inverse. Note that may not be complete inverse of because the composition in the other order fg may not be the identity on Y .In other words, a function that can be undone or reversed, such as is not necessarily invertible (bijective). Injections are reversible but not 7

always invertible. Although it is impossible to reverse a non – injective (and therefore information losing) function, you can at least obtain a quasi of it that is a multiple valued function. Injection may be made invertible: 3

In fact to change an injective function f:  X Y into a bijective (here invertible) function, it suffices to replace its codomain Y by its actual range j f X .That is, let g: X J such that g x  f xfor all x in X; then J is injective. Indeed, f can be

factored as ()incljy, g where incll j,y is the inclusion function from J into Y. Other properties: i. If f and g are both injective, then fg is injective. ii. f: X Y is injective if and only if, given any functions g,h: W X whenever f g f h,then gh . In other words injective functions are precisely the monomorphism in the category set of sets. iii. If is an injective and A is a subset of X, then f1  f A  A. Thus, A can be recovered from its image fA . iv. If is injective and A and B are both subsets of X, then f()()() A B f A f B v. Every function h: W Y can be decomposed as fgfor a suitable injection f and surjectiong . This decomposition is unique up to isomorphism, and may be thought of as the inclusion function of the range hW of h as a subset of the codomain y of . vi. If f: X Y is an injective function, then Y has at least as many elements as X , in the sense of cardinal numbers.

vii. If both X and Y is finite with the same number of elements, then f: X Y is injective if and only if f is subjective. viii. Every embedding is injective. 3

4 The bijective function: 1.2.2Definition : In mathematics a bijection, or a bijective function is a function f from a set to a set Y with a property that, for every y inY , there is exactly one x in such that f(x) = y.

 X  Y 

8 32

12 48

Alternatively, is a bijective if it is a one – to –one correspondence between those sets; i.e., both one- to – one (injective) and onto (surjective) (see in this text, bijection, injection and surjection).For example, consider the function sumdif (sum- difference) that to each pair xy, of real numbers associates the pairx, y  ( x  y ,  x – y ).A bijective function is also called a permutation. This is more commonly used when XY . It should be noted that one – to –one function means one – to – one correspondence (i.e., bijection).To some authors, but injection to others. The set of bijections from to is denoted as XY Bijective functions play a fundamental role in many area of mathematic, for instance in the definition of isomorphism, permutation group, projective map, and many others.

4 Composition and inverses:

A function f is bijective if and only if its inverse relation f 1is a function the inverse of gf is()()()g f1 f  1 g  1 . In the other hand, if the composition gf of two functions is bijective, we can only say that is injective and g is surjective. A relation from X to Y is an objective function if and only if there exist another relation g from Y to that gfis the identity function on , and fgis the identity function on . Consequently, the set has the same cardinality. The following diagram shows a bijection composed of an injection and surjection

( ) ( ) (Z)

1 D P

2 B Q

3 C R

Examples and counter examples:

i. For any set , the identity function id X from to , defined

by idX  x, is a bijeticve. ii. The function from a real line Rto defined by f x 21 x is a bijective, since for each y there is a unique xy 12 such that f x  y iii. The exponential function g:, R R with g() x ex , is not bijective :for instance there no x in such that gx  1,showing that g is not surjective, however if the

co-domain is change to be the positive real number R 0, then g becomes bijective ;its inverse is the natural logarithm function Ln.

4 Properties:

i. A function f from the real line R to is bijective if and only if its plot is intersected by any horizontal line at exactly one point. ii. if X is a set, then the bijective functions from to itself, together with the operation of functional composition , form a group, the symmetric group of which is denoted by

SXS , ix, or X ! iii. For a subset Aof the domain and subset Bof the co-domain we have: f() A A And f1 B B . iv. If and Y are finite sets with the same cardinality, and f:. X Y then the following are equivalent: a) Is a bijection. b) Is a surjection. c) Is injection. d) At least for a finite set S ,there is a bijection between the set of possible total ordering of the elements and the set of bijections from to . That to say, the number of permutations (another name for the bijection) of elements of is the same as the number of total orderings of the set –namely, n!

The surjection function: 5

1.2.3Definition : In mathematics, a function is said to be surjective if its values spans its whole codomain; that is, for every y in the codomain, there at least one x in the domain such that f x  y .

Said another way, a function is a surjective if and only if its range fx()is equals to its codomainY . A surjective function called surjection, and also said to be onto. The following diagrams 1 and 2 shows surjection

X 1 2 1 1 D D

2 2 B B 3 3 C

C 4 4 A

A surjective function 1,another surjective (this one happens to be bijection

3

1 B

C 2 A 3

A non – surjective function (this one happens to be an injection3 Examples and counterexamples: 5

i. For any set , the identity function idx on is surjective.

ii. The function f: R R defined by f x 21 x is surjective (and even bijective), because for every real number y we have f x  ywhere xy  –1 2 iii. The natural logarithm function Ln :(0, )is surjective. iv. The function fZ: 0,1,2,3 defined by f x  xmode 4 is a surjective. v. The function g: R R defined by g x  x2 is not surjective, because (for example there is no real number such that x  . However, if the co-domain is defined as0,, then g is surjective.

There is always exists a function “reversible” by a surjection: 5

Every function with a right inverse is a surjection. The converse is equivalent to the axiom of choice. That is, assuming choice, a function f: X Y is surjective if and only if there exists a function g: Y X such that, for every ( g can be undone by f ). That is a function g such that fgequals the identity function on Y. Note that g may not be a complete inverse of because the composition in the other order gf may not be the identity on X. In other words, can undo by or “reverse” , but not necessarily can be reversed by it. Surjections are not always invertible (bijective). Surjective composition: The first function need not be surjective

X Y Z

1 B P 2

C Q 3 4 A R

Other properties: 5

i. If f and g are both surjective, then fgis surjective. ii. If is a surjective, then f is surjective (but g may not be) iii. f: X Y Is surjective if and only, given any function g,: h Y Z , whenever g f h f then gh . In other words surjective functions are precisely the epimorphism in the category set of sets. iv. If f: X Y is surjective and Bis a subset ofY , then f( f1 ( B ))  B, thus B can be recovered from its preimage fB1(). v. For any function h: X Z there exists a surjection f: X Y and injection g: Y Z such thath g f . To see, this defines Y to be the sets hz1()where z is inZ . These sets are disjoint and portions’ X . Then, f carries each to the element of Y which contains it, and g carries each element of Y to the point in to which h sends its points. Then is surjective it is the projection map, and g is injective by the definition. vi. By collapsing all arguments mappings to a given fixed image, every surjection induces a bijection defined on a quotient of its domain. More precisely, every surjection f: A B can be factored as a projection followed by a bijection as follows. Let A : be the equivalence classes of Aunder the following equivalence relation: xy: if and only if f x  f() y . Equivalently, is the set of all preimages under let PAA:::  be the projection map which sends each x in

to its equivalence classx, and let fp : A A : be the well-

defined function given by fp : A:  B be the well defined

function given by fp  x  f( x ). Then fPp : . vii. If f: X Y is a surjective function, then X has at least as many element asY , in the sense cardinal numbers. viii. If both and is finite with the same number of elements, then f: X Y is surjective if and only if is surjective. 5

The isometry: 6

In mathematics, an isometry, isometric, isomorphism or congruence mapping is the distance preserving isomorphism between metric spaces. Geometric figures which can be related by isometry are called congruent.

1.2.4Definitions:

a) Let X and Y be metric spaces with metric dx andd y .A map f: X Y is called a distance preserving if for any x, yε X one

hasdyx f x,(,) f y  d x y .Distance preserving map is automatically injective. b) A global isometry is a bijective distance preserving map. c) A path isometry or arc wise isometry is a map which preserves the lengths of the curves (not necessarily bijective) d) Two metric spaces X and are called isometric if there an isometry from to The set of isometrics from a metric space to itself form a group with respect to function composition, called the isometry group.

Examples:

i. Any reflection, translation or rotation is a global isometry on Euclidean spaces. ii. The map RR defined by xx is a path isometry. iii. The isometric linear maps fromCn to itself are the unitary matrices.

Linear isometry:

1.2.5Definition:

Given two normed metric spaces V andW , a linear isometry is a linear map f: V W that preserves the norms:

f() v v For all v inV Linear isometric are distances – preserving maps in the above sense. They are global if and only if they are surjective.

Generalization: 6 Given positive real numberε an ε -isometric or almost isometry (also called Haussdroff approximation) is a map f: X Y between metric spaces such that: ι ιι i. For x, xε X one has dYX( f ( x ), f ( x )) d ( x , x ) <ε, and ii. For any point yYε there exists a point xXε

withdY  y, f x<εdY . That is an ε -isometry preserves distances to within  and leaves no element of the co-domain further than ε away from the image of an element of the domain. Note that - isometrics are not assumed to be continuous. Quasi isometry is yet another useful generalization. 6

The homomorphism: 7

1.2.6Definition: In abstract algebra, a homomorphism is a structure- preserving map between two algebraic structures (such as groups, rings, or vector spaces).The word homomorphism comes from Greek language: homo meaning “same” and morphe meaning “shape”.

Examples:

i. Consider the natural numbers with addition as operation. A function which preserves addition should have this property: f a b  f a  f b And for the function f(x) = 3x we have f a b 33 a  b  f a  f b Note that this homomorphism maps the natural numbers back into themselves.

ii. 2-Homomorphisms do not have to map sets which have the same operations. For example operation preserving function exist between the set of real numbers with addition and the set of positive real numbers with multiplication. A function which preserves this operation should have this property: f()()() a b  f a  f b , since addition is the operation on the first set and multiplication on the second. Given the laws of exponents, f() x ex satisfies this condition: 2 + 3 = 5translate intoe2 e 3 e 5 . iii. A particularly important property of homomorphism is that if an identity element is present, it is always preserved, that is, mapped to the identity. In the first example f 00  , and in the second example, f 01  since 0 is the additive identity and 1is the multiplicative identity.

1.2.7Formal definition: 7 A homomorphism is a map from one algebraic structure to another of the same type that is preserves all the relevant structure; i.e. properties like identity elements, inverse elements, and binary operations.

Types of homomorphism:

i. An isomorphism is a bijective homomorphism. ii. An epimorphism is a surjective homomorphism. iii. A homomorphism from an object to itself is called an endomorphism. iv. An endomorphism which is also an isomorphism is called an automorphism.

Kernel of a homomorphism: 7

Any homomorphism f: X Y defines an equivalent relation ~ on X by ab: if and only if f a  f b.The relation ~ is called the kernel of f . It is a congruence relation on . The quotient set X : can be given an object structure in natural way, i.e.x y  x  y. In that case the image of in Y under the homomorphism is necessarily isomorphic to : this fact is one of the isomorphism theorems.

8 13 The equivalence relation:

In mathematic, an equivalence relation is, loosely, a binary relation on a set that specifies how to split up (i.e.partitions) the set into subsets such that every element of the larger set is in exactly one of the subsets. Any two numbers of the larger set are then considered “equivalent” with respect to the equivalence relation if and only if they are element of the same subset. Notation:

Although various notations are used throughout the literature to denote that two elements a and bare equivalent with respect to equivalence relation R, the most common are“ab ~ ”and""ab , which are used when is the obvious relation been referenced, and variation of “ab ~R ”, “abR ”,, or “aRb ”. 1.3.1Definition:

Let Abe a set and ~ be a binary relation on A. ~ is called an equivalence relation if and only if for all a,, b cε A, all the following holds true: i. Reflexivity: a ~ a ii. Symmetry: a ~ b then b ~ a iii. Transitivity: if a ~ b and b ~ c then a ~ c.

The equivalence class of a under ~, denoted a,is defined as a   bε A a: b . A with ~ is called setoid.

8 Examples:

Equivalence relations:

i. The equality relation (“=”) between elements of any set. ii. “Is similar to” or “congruent to” in the set of triangle. iii. Let abc,,be natural numbers, and let ab, and cd, be ordered pairs of such numbers. Then the equivalence classes under the relation a,, b :  c d are the: * Integers ifa d  b  c, * Positive rational numbers ifad cd . iv. Greens relations: are five equivalence relations on the elements of semigroups that characterize the elements of semigroups in terms of the principal ideals they generate.

Relations that not equivalence:

i. The relation " "between real numbers is reflexive and transitive but not symmetric. For example, 7 ≥ 5 does not imply that 5 ≥ 7 ii. The concept of parallelism in ordered geometry is not symmetric and, is therefore, not an equivalence relation.

1.3.2Definitions:

Let X be a non empty set, and leta, bε X . These are some definitions:

Equivalence class: the set of all a and b for which a ~ b holds make up an equivalence class of X by ~. Leta   xε X x: adenote the equivalence class to which a belongs. Then all the elements of X equivalent to each other are also elements of the same equivalence class. 19

Quotient set: the set of all possible equivalence classes of X by ~, denoted X: :   X xε X , is the quotient set of X by ~. If X is a topological space, there is a natural way of transforming X : into a topological space;

Projections: the projection of ~ is the function π : XX defined by π()xx   which maps elements of X into their respective equivalence classes by~. 8 1.3.1 Theorem on projection: Let the function f: X B be such that a: b f a f b then there is a unique function g:, X:  B such that fg π. If f is a surjection and a: b f a f b. Then g is a bijection.

Equivalence kernel:

1.3.3Definition:

The equivalence kernel of a function f is the equivalence relation ~ defined by x: y f()() x f y . The equivalence kernel of an injection is the identity relation.

Partition:

1.3.4Definition: a partition of X is a set P of subsets of X, such that every element of X is an element of a single element of P. Each element of P is a cell of the partition. Moreover, the elements of P are pairwise disjoint and their union is X.

1.3.2 Fundamental theorems of equivalence relations:

i. An equivalence relation ~ on a set X partitions X ii. Conversely, corresponding to any partition of X, there exists an equivalence relation ~ on X.

9 Counting possible partition:

The number of possible equivalence relations on X equals the number of distinct partition of X, which is the nth Bell number:  n Bn   k ek! kn

8 Generating equivalence relation:

i. Given any set X, there is an equivalence relation over the set XX of all possible functions XX .Two such functions are deemed equivalent when their respective sets of fixpoints have the same cardinality, corresponding to cycles of length one in the permutation. Functions equivalent in this manner form an equivalent class on and these equivalence classes Partition . ii. An equivalence relation~ on X is the equivalence kernel of its surjective projectionπ : XX : .Conversely, any surjection between sets determine a partition on its domain, the set of preimages of singletons in the co-domain. Thus an equivalence relation over X, a partition of X, and a projection whose domain is X, are three equivalent ways of specifying the same thing. iii. The intersection of any collection of equivalence relations over X (viewed as a subset of XX ) is also an equivalence relation. iv. E quivalence relations can construct new spaces by “gluing things together”. Let X be the unit Cartesian square 0,1 0,1and let ~ be the equivalence relation on X defined bya, bε  0,1 a , o :  a ,1 (0, b )(1, b ).Then the quotient space X : can be naturally identified with a tours: take a square piece of paper, bend and glue together the upper and lower edge to form a cylinder, then bend the resulting cylinder so as to glue together its two open ends, resulting a tours.

8 Algebraic structure:

Much of mathematics is grounded in the study of equivalences and order relations. It very well known that lattice theory captures the mathematical structure of order relations. Even though equivalence relations are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is not as well known as that of orders. The former structure draws primarily on group theory and, to lesser extent, on the theory of lattices, categories, and groupoids.

1.3.3Group theory:

Just as order relation are grounded in ordered sets, sets closed under supremum and infimum, equivalence relations are grounded in partitioned sets, sets closed under bijections preserving partition structure. Since all such bijections map an equivalence class onto itself, such bijections are also known as permutations. Hence permutations groups (transformation groups) and related notation of orbit shed light on the mathematical structure of equivalence relations. Let “~” denote an equivalence relation over some nonempty set A, called the universe or underlying set. Let G denote the set of bijective functions over A that preserve the partition structure of A:, xε A g ε G g x ε  X  then the following three connected theorems holds: i. ~ partitions A into equivalence classes. ii. Given a partition of A, G is a transformation group under composition, whose orbits are the cells of the partitions. iii. Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classes are the orbits of G. In sum, given an equivalence relation ~ over A, there exist a transformation group G over A, whose orbits are the equivalence classes of A under ~. The transformation group characterization of equivalence relations differs fundamentally from the way lattices characterize order

22 relations. The arguments of the lattice theory operations meet and join are elements of some universe A. Meanwhile, the arguments of the transformation group operation group operations composition and inverse are elements of a set of bijections, AA .Moving to groups in general, let H be a subgroup of some group G, let ~ be an equivalence relation on G, such thata: b ab1ε H . The equivalence classes of ~ also called the orbits of the action of H on G are the right cosets of H in G. Interchanging a and b yields the left cosets.

8 Categories and groupoid:

The composition of morphisms central to category theory, denoted here by concatenation generalizes the composition of functions central to transformation groups. The axioms of category theory assert that the composition of morphisms associates, and that the left and right identity morphisms exist for any morphism. If a morphism f has an inverse, f is an isomorphism, i.e., there exists a morphism g such that the compositions fg and gf equal the appropriate identity morphisms. Hence the category theoretic concept nearest to an equivalence relation is a (small) category whose morphisms are all isomorphisms. Groupoid is another name for small category of this nature. Let G be a set and let “~” denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows: The objects are the elements of G, and for any two elements x and y of G, there exist a unique morphism from x to y if and only if x ~ y. There may be many such g, each of which can be regarded as a distinct “proof” that x and y are equivalent. The advantages of regarding an equivalence relation as special case of groupoid include: i. Whereas the notation of free equivalence relation does not exist, that of free groupoid on a directed graph does. Thus it is meaningful to speak of a “representation of an equivalence

relation, i.e., a representation of the corresponding groupoid. Bundles of groups, group action, sets, and equivalence relations can be regarded as special cases of notation of groupoid, a point of view that suggest a number of analogies; ii.In many context “quotiening,” and hence the appropriate equivalence relations often called congruence’s, are. This leads to the notation of an internal groupoid in the category.

8 Lattices:

The possible equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called Con X by convention. The canonical mapker : XXˆ  ConX , relate the monoid. XXˆ Of all functions on X and Con X .Ker is surjective but not injective. Less formally, the equivalence relation Ker on X, takes each function f: X X to its kernel Ker f. Likewise, Ker (Ker) is equivalence relation on

Equivalence relations and mathematical logic:

Relations are a ready source of examples and counter examples. For example, an equivalence relation with exactly two Infinite equivalence classes are an easy example of a theory which is ω -categorical, but not categorical for any larger cardinal number. An implication of model theory is that the properties defining a relation can be proof independent of each other (and hence necessary parts of the definition) if and only if, for each property , examples can be found of relations not satisfying the given property while satisfying all the other properties. Hence the three defining properties of

24 equivalence relations can be proof mutually independent by the following examples: i. Reflexive and transitive: The relation ≤ on N. Or any preorder; ii. Symmetric and transitive: The relation R on N, define as aRb ab 0. or any partial equivalence relation; iii. Reflexive and symmetric: The relation R on Z, defined as aRb" a bis divisible by at least one of 2 or 3”. Or any dependency relation. Properties definable in first order logic that an equivalence relation may or may not possess include: i. The number of equivalence classes is finite or infinite; ii. The number of equivalence classes equals the (finite) natural number π; iii. All equivalence classes have finite cardinality; iv. The number of elements in each equivalence class is a natural 8 9 number n.

Euclid anticipated equivalence:

Euclid’s the elements includes the following “Common notation 1”: Things which equals the same thing also equal one another. Nowdays, the property described by common notation 1 is called Euclidean (replacing “equal” by “are in relation with”). The following theorem connects Euclidean relations and equivalence Relations:

1.3.4 Theorem: If a relation is Ecluidean and reflexive, it is also symmetric and transitive.

Proof: i. aRc bRc  aRb a\() c  aRa  bRa  aRb[reflexive relation ,eraseT ] bRa aRb. Hence R is symmetric. ii. ()().aRc bRc  aRb symmetric  aRc  cRb  aRb Hence R is transitive. Hence an equivalence relation is a relation that is Euclidean and reflexive. The elements mentions neither symmetry nor reflexivity, and Euclid probably would have deemed the reflexivity of equality too obvious to warrant explicit mention. If this (and taking “equality” as an all – purpose abstract mention) is granted, a charitable reading of Common Notation 1 would credit Euclid with being the first to conceive of equivalence relations and their importance in deductive 8 Systems

CHAPTER TWO

VECTOR-VALUED SINGULAR INTEGRALS

We start our chapter with some questions and their answers Firstly we ask what is the importance of the singular integrals? Answer: various important operations in mathematics and mathematical sciences are described by integrals which do not converge absolutely but nevertheless have a meaning in the principal value or other weak sense. The systematic study of the existence of such singular integrals and their boundedness properties is known as the Calderón-Zygmund Operator (CZO), Which has been the object of intensive study since its origin in the 50's. ( An approach to singular integrals is the Hilbert transform in extending results involving Hilbert operator to cases involving functions defined on Euclidean spaces Rn , n >1 Calderón and_Zygmund considered some classes of convolutions operators whose kernels although having specified 23 singularities satisfy various smoothness conditions) Secondly we ask why the vector valued functions? Answer: In the last 25 years, there has been much interest towards a vector-valued extension of Calderón-Zygmund theory, which would allow the estimation of singular integrals of functions taking values in finite-dimensional Banach space. Such a need has come especially from applications in the theory of partial differential equations, which are not limited to the Hilbert space frame work but also ask for a treatment of more general Banach spaces. (The theory of singular integrals has half century background of intensive development. The main ingredient lemma appeared in 1952 by Calderón-Zygmund theory, in 1960 Hörmmander introduced his cancellation conditions which admit boundedness extension from the space Lp to , 1< p < and from space H1 to L1 and from space to L ( weak ) nowadays the adjoint operator bounded

27 from L to BMO (bounded mean osciliator).In 1972 C.L.Fefferman and E.M.Stien prove HL11 and L  BMOboundedness of (CZO). R .Coifman obtains at 1984 the HHpp boundedness of (CZO) of the case of one dimension. Many authors have important contributes in these 10 extensions in the recent years) (There have been several directions in which these spaces been extended and developed one of them the consideration of convolution operator with kernels similar in higher dimensions It took time to isolate the properties that were needed in the kernel in order to apply similar technique. After some preliminary stages in the case of convolution kernels, the theory 13 Extended to non convolution operators). Thirdly we ask how we obtain these extensions? Answer: In order to obtain such an extension, new methods using probabilistic concepts, especially so-called unconditional martingale differences (UMD), were developed by D. L.Burkholder, J. Bourgain, T. R. McConnell and T. Figiel in the 80's. More recently, the formalization and systematic use of the notion of randomized boundedness or R-boundedness implicit in the work of Bourgain has given new boost to the field. Roughly speaking, unconditionality becomes the substitute for the notion of orthogonality, which only exists in Hilbert spaces, and R- boundedness is a stronger version of boundedness, which one needs to compensate for the fact that the substitute is not quite the same as the original. This talk is intended as an introduction to the main ideas, notions and working tools in the field of vector-valued singular integrals, which is an interesting meeting place of harmonic, functional and stochastic analysis

 21   The vector space:

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied ("scaled") by numbers, called scalars . An additional datum is an order ≤, a token by which vectors can be compared.

28 Vector spaces are the subject of linear algebra and are well- understood from this point of view, since vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. The theory is further enhanced by introducing on a vector space some additional structure, such as a norm or inner product. Such spaces arise naturally in mathematical analysis, mainly in the guise of infinite-dimensional function spaces whose vectors are functions. Analytical problems call for the ability to decide if a sequence of vectors converges to a given vector. This is accomplished by considering vector spaces with additional data, mostly spaces endowed with a suitable topology, thus allowing the consideration of proximity and continuity issues. These topological vector spaces, in particular Banach spaces and Hilbert spaces, have a richer theory. Today, vector spaces are applied throughout mathematics, science and engineering. They are the appropriate linear algebraic notion to deal with systems of linear equations, offer a framework for Fourier expansion, which is employed in image compression routines, or provide an environment that can be used for solution techniques of partial differential equations. 

22 Topological vector spaces: 12

Banach spaces: 23 2.2.1Definition:

Banach spaces are defined as complete normed vector spaces. This means that a Banach space is a vector space V over the real or complex numbers with a norm  such that every Cauchy sequence (with respect to the metric d x, y  x y in has a limit in . Since the norm induces a topology on the vector space, every Banach space is necessarily metrizable and metrizable spaces generally have very interesting properties. A first example is the vector space p consisting of infinite

29 vectors with real entries X  x12, x ,.... whose p-norm 1p  given by

1 p p xxp :  i For p < and xx : supii 2 2 1 i

is finite.

The topologies on the infinite-dimensional p are inequivalent for different p. E.g. the sequence of vectors n  n  n n Xn  (  ,  ,  ...,  , 0, 0, ...),i.e. the first  components are the following ones are 0, converges to the zero vector for nn p , but does not for p 1: xn  sup 2 ,0  2  0 , n 2 n n n but xn 1  2  2  2  1 i1 More generally than sequences of real numbers, functions fR: are endowed with a norm that replaces the above sum by the Lebesgue integral

p 1 f: f x dx p 222 p        The space of integrable functions on a given domain  (for example an interval) satisfying f p <, and equipped with this norm are called Lebesgue spaces, denoted Lp  .These spaces are complete. (If one uses the Riemann integral instead, the space is not complete, which may be seen as a justification for Lebesgue's integration theory) .Concretely this means that for any sequence of Lebesgue-integrable functions

ƒ ,ƒ , ...with fn p <, satisfying the condition

p limf x  f x dx 0 2 2 3 kn,   k n  

There exists a function fx  belonging to the vector space such that

30 p limf x  f x dx 0 224 k  k  

Imposing boundedness conditions not only on the function, but also on its derivatives leads to Sobolev spaces.

Hilbert spaces: 23 12

2.2.2Definition:

Complete inner product spaces are known as Hilbert spaces, in honor of David Hilbert. The Hilbert space L2  , with inner product given by

f g  f x g x dx 2 2 5

Where gx  denotes the complex conjugate of gx  . Is a key case in harmonic analysis.

Sobolev space: 14

2.2.3Definition:

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space. Intuitively, a Sobolev space is a Banach space or Hilbert space of functions with sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and smoothness of a function. Their importance lies in the fact that solutions of partial differential equations are naturally in Sobolev spaces rather than in the classical spaces of continuous functions and with the derivatives understood in the classical sense. With this definition, the Sobolev spaces admit a natural norm,

31 1/pp 1/ kk ()()ip   i p  f f| f ( t )| dt . 2 2 6 kp,       ii00p   

Space W kp, equipped with the norm  is a Banach space. It kp, turns out that it is enough to take only the first and last in the sequence, i.e., the norm defined by

ff()k  2 2 7 p p

Is equivalent to the norm above

–  Kernel (mathematics): 15

In mathematics, the word kernel has several meanings. Kernel may mean a subset associated with a mapping:

i. The kernel of a mapping is the set of elements that map to the zero element (such as zero or zero vector), as in kernel of a linear operator and kernel of a matrix. More generally, the kernel in algebra is the set of elements that map to the neutral element. Here, the mapping is assumed to be a homomorphism, that is, it preserves algebraic operations, and, in particular, maps neutral element to neutral element. The kernel is then the set of all elements that the mapping cannot distinguish from the neutral element. ii. The kernel in category theory is a generalization of this concept to morphisms rather than mappings between sets. iii. In set theory, the kernel of a function is the set of all pairs of elements that the function cannot distinguish, that is, they map to the same value. This is a generalization of the kernel concept above to the case when there is no neutral element. iv. In set theory, the difference kernel or binary equalizer is the set of all elements where the values of two functions coincide.

32 Kernel may also mean a function of two variables, which is used to define a mapping:

i. In integral calculus, the kernel (also called integral kernel or kernel function) is a function of two variables that defines an integral transform, such as the function k in

(Tf )( x ) X k ( x , x ) f ( x  ) dx  . 2 3 1

ii. In partial differential equations, when the solution of the equation for the right-hand side f can be written as Tf above, the kernel becomes the Green's function. The heat kernel is the Green's function of the heat equation. iii. In the case when the integral kernel depends only on the difference between its arguments, it becomes a convolution kernel, as in

(Tf )( x )X  ( x x ) f ( x  ) dx  . 232 iv. In probability theory and statistics, stochastic kernel is the transition function of a stochastic process. In a discrete time process with continuous probability distributions, it is the same thing as the kernel of the integral operator that advances the probability density function. v. Kernel trick is a technique to write a nonlinear operator as a linear one in a space of higher dimension. vi. In operator theory, a positive definite kernel is a generalization of a positive matrix. vii. The kernel in a reproducing kernel Hilbert space.

Kernel (linear operator): 15 2.3.1Definition:

In linear algebra and functional analysis, the kernel of a linear operator L is the set of all operands v for which Lv  0. That is, if LVW:  , then

33 ker(L ) v  V : L ( v )  0 , 2 3 3

Where 0 denotes the null vector inW . The kernel of L is a linear subspace of the domainV .The kernel of a linear operator RRmn is the same as the null space of the corresponding nm matrix. Sometimes the kernel of a general linear operator is referred to as the null space of the operator.

Properties: 15

If LVW:  , then two elements of V have the same image in W if and only if their difference lies in the kernel of L: L( v ) L ( w )  L ( v  w )  0. 234

It follows that the image of L is isomorphic to the quotient ofV by the kernel: im(LVL ) / ker( ). 2 3 5

When V is finite dimensional, this implies the rank-nullity theorem: dim(kerLLV ) dim(im ) dim( ). 2 3 6

When V is an inner product space, the quotient VLker  can be identified with the orthogonal complement in V ofker L. This is the generalization to linear operators of the row space of a matrix.

If V and W are topological vector spaces (andW is finite- dimensional) then a linear operator LVW:  is continuous if and only if the kernel of L is a closed subspace of V (5)

Kernel trick: In machine learning, the kernel trick is a method for using a linear classifier algorithm to solve a non-linear problem by

34 mapping the original non-linear observations into a higher- dimensional space, where the linear classifier is subsequently used; this makes a linear classification in the new space equivalent to non-linear classification in the original space. This is done using Mercer's theorem, which states that any continuous, symmetric, positive semi-definite kernel function K x, y can be expressed as a dot product in a high-dimensional space. More specifically, if the arguments to the kernel are in a measurable space X , and if the kernel is positive semi-definite i.e.

 K( xi , x j ) c i c j  0 2 3 7 ij,

For any finite subset xx1,...., nof X and any real numbers

CC1,...., nthen there exists a function   x whose range is in an inner product space of possibly high dimension, such that K( x , y ) ( x )g ( y ). 2 3 8

The kernel trick transforms any algorithm that solely depends on the dot product between two vectors. Wherever a dot product is used, it is replaced with the kernel function. Thus, a linear algorithm can easily be transformed into a non-linear algorithm. This non-linear algorithm is equivalent to the linear algorithm operating in the range space of . However, because kernels are used, the function is never explicitly computed. This is desirable, because the high-dimensional space may be infinite- dimensional (as is the case when the kernel is a Gaussian). 6

2.3.1Calderón-Zygmund kernels theory: 18

A function KRRR: nn is said to be Calderón-Zygmund kernel if it satisfies the following conditions for some constants C > 0 and  > 0

35 a C K x, y   n 2 3 9 xy

 C x x b  2 3 10 K x,, y  K x y  n  x y  x  y 

Whenever x x 1 max x  y , x  y 2  

 C y y  c K x,, y  K x y  n 2 3 11  x y  x  y  1 Whenever yy maxx y , x y 2  

Poisson kernel: 24

In potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Dirichlet problem. Specifically, it gives solutions to the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disc. The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson. The Poisson kernel is important in complex analysis because its integral against a function defined on the unit circle , the Poisson integral , gives the extension of a function defined on the unit circle to a harmonic function on the unit disk. By definition, harmonic functions are solutions to Laplace's equation, and, in two dimensions, harmonic functions are equivalent to meromorphic functions. Thus, the two- dimensional Dirichlet problem is essentially the same problem as that of finding a meromorphic extension of a function defined on a boundary. Poisson kernels commonly find applications in control theory and two-dimensional problems in electrostatics. In practice, the definition of Poisson kernels are often extended to n-dimensional problems.

36 Two-dimensional Poisson kernels: 24

On the unit disc:

In the complex plane, the Poisson kernel for the unit disc is given by

11r2  rei  P( ) r||n ein   Re , 0  r  1. r  2 i n 1 2r cos  r 1  re 2 3 12

This can be thought of in two ways: either as a function of r and , or as a family of functions of  indexed byr .If D  z : z 1is the unit disc in C and if f is a continuous function from the unit circle Dinto Rthen the function u given by

u( rei )1  P (  t ) f ( e it ) dt , 0  r  1 2 3 13 2  r   Or equivalently by

1  ezit  u( z ) Re f ( eit ) dt 2 3 14  it   2 ez

Is harmonic in D and extends to a continuous function on D that agrees with f on the boundary of the disc. It is common to restrict oneself to functions which are either square integrable or p-integrable on the unit circle. When one also asks for the harmonic extension to be holomorphic, then the solutions are elements of Hardy space. In particular, the Poisson kernel is commonly used to demonstrate the equivalence of the Hardy spaces on the unit disk, and the unit circle. In the study of Fourier series the Poisson kernel arises in the study of Abel

37 means for a Fourier series, and gives an example of a summability kernel (Katznelson 1976)

On the upper half-plane: 24

The unit disk may be conformally mapped to the upper half- plane by means of certain Möbius transformations. Since the conformal map of a harmonic function is also harmonic, the Poisson kernel carries over to the upper half-plane. In this case, the Poisson integral equation takes the form

uxiy()()() 1  Pxtftdt  2 3 15   y   For y 0. The kernel itself is given by y Pxy(). 2 3 16 xy22

Given a function f Lp() R , the Lp space of integrable functions on the real line, then u can be understood as the harmonic extension of f into the upper half-plane. In analogy to the situation for the disk, when u is holomorphic in the upper half- plane, then u is an element of the Hardy spaceuH p , and, in particular,

ufH p  Lp 2 3 17

Thus, again, the Hardy space H p on the upper half-plane is a Banach space, and, in particular, a closed subspace of LRp(). The situation is only analogous to the case for the unit disk; the Lebesgue measure for the unit circle is finite, whereas that for the real line is not.

On the ball:

n For the ball of radius rB,,r in R the Poisson kernel takes the form

38 22 rx| | Px(,)  n 2 3 18 rxn1 ||

Where xB ,  S (the surface of B ), and  is the surface r r n1 area of the unit sphere. Then, if ux  is a continuous function defined on S, the corresponding Poisson integral is the function P u xdefined by

P u x   u P ,  d    2 3 19 S

It can be shown that P u x is harmonic on the ball and that P u x extends to a continuous function on the closed ball of radius r,and the boundary function coincides with the original function u.

On the upper half-space: 24

An expression for the Poisson kernel of an upper half-space can also be obtained. Denote the standard Cartesian coordinates of Rn1 by

(,)(,,,)t x t x1 xn 2 3 20

The upper half-space is the set defined by Hnn11{( t ; X )  R t  0} 2 3 21

The Poisson kernel for H n1 is given by t P(,) t x cn 2 3 22 (tx22 | | )(n 1)/2

Where [(n 1)/ 2] cn  . 2 3 23  (n 1)/2

39 The Poisson kernel for the upper half-space appears naturally as the Fourier transform of the Abel kernel

K t,   e 2t | | 2 3 24

In which t assumes the role of an auxiliary parameter to it,

2t |  | 2  i  · x P( t , x )F ( K ( t ,·))( x ) n e e d . 2 3 25 R

In particular, it is clear from the properties of the Fourier transform that, at least formally, the convolution P[ u ]( t , x ) [ P ( t ,·) u ]( x ) 2 3 26

Is a solution of Laplace's equation in the upper half-plane. One can also show easily that as t  0, P u t , x u x in a weak sense.

– Convolution: 16

Visual explanation of convolution:

1. Express each function in terms of a dummy variable . 2. Transpose one of the functions: gg  .

40 3. Add a time-offset,t which allows gt   sliding along the  -axis. 4. Start  at  and slide it all the way to. Wherever the two functions intersect, find the integral of their product. In other words, compute a sliding, weighted-average of function f   , where the weighting function is g  . The resulting waveform (not shown here) is the convolution of functions f and g . If ft  is a unit impulse, the result of this process is simply gt  which is therefore called the impulse response. 2.4.1Definition: 16

The convolution of f and g is written fg . It is defined as the integral of the product of the two functions after one is reversed and shifted. As such, it is a particular kind of integral transform:

  f g  f  g t   d    f t    g  d  2 4 1

More generally, if f and g are complex-valued functions on Rd , then their convolution may be defined as the integral: def  fgx  RRdd fygxydy     fxygydy     . 242

Compactly supported functions:

If f and g are compactly supported continuous functions, then their convolution exists, and is also compactly supported and continuous (Hörmander). More generally, if either function (say f ) is compactly supported and the other is locally integrable, then the convolution fg is well-defined and continuous

41 16 Integrable functions:

The convolution of f and g exists if and are both Lebesgue integrable functions (in LR1 d  ), and fg is also integrable (Stein & Weiss 1971, Theorem 1.3). This is a consequence of Tonelli's theorem. Likewise, if f L1 Rd  and g Lp  Rd  where1p , then f g Lp  Rd  and

f gp f1 g p . 2 4 3

More generally, Young's inequality implies that the convolution is a continuous bilinear map between suitable Lp spaces. Specifically, if 1pqr , , satisfy

1 1  1 1, 244 p q r  

Then

pq f*,,, grp f gq f  L g  L 2 4 5

So that the convolution is a continuous bilinear mapping from LLpq to Lr .

–  Integral transform: 17

2.5.1Definition:

In mathematics, an integral transform is any transform T of the following form:

t Tf()(,)(). u  2 K t u f t dt 2 5 1 t1

The input of this transform is a function f , and the output is another functionTf . An integral transform is a particular kind of

42 mathematical operator. There are numerous useful integral transforms. Each is specified by a choice of the function K of two variables, the kernel function or nucleus of the transform. Some kernels have an associated inverse kernel K1 u, t  which (roughly speaking) yields an inverse transform:

u f( t ) 2 K1 ( u , t )( Tf ( u )) du . 2 5 2 u1

A symmetric kernel is one that is unchanged when the two variables are permuted. 17

–  Singular integral: 18

In mathematics, singular integrals are central to abstract harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator

T f x   K x, y f y dy 2 6 1

Whose kernel function KRRR: n n n is singular along the diagonal xy . Specifically, the singularity is such that K x, y is of size xy n asymptotically as xy0. Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over xy > as 0, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on LRp  n  .

43 The Hilbert transform: 18

2.6.1Definition:

The archetypal singular integral operator is the Hilbert transform H . It is given by convolution against the kernel K x 1  x for x in R. More precisely,

1 1 H f x  lim ||xy f y dy. 2 6 2   xy The most straightforward higher dimension analogues of these are the Riesz transforms, which replace K x 1 x with x Kx   i 2 6 3 i x n1

n Where in1,...., and xi is the i th component of x in R . All of these operators are bounded on Lp and satisfy weak-type (1,1) estimates.

Singular integrals of convolution type:

2.6.2Definition:

A singular integral of convolution type is an operator T defined by convolution with a kernel K that is locally integrable on Rn \0  , in the sense that

Txf  lim K 2 6 4     ||xy xy f y dy.  

Suppose that the kernel satisfies:

1. The size condition on the Fourier transform of K supR| x | 2 R |K ( x )| dx C , 2 6 5 R0

44 2. The smoothness condition: for someC 0, sup xy2 K x y  K x dx  C . 266 y0

And

3. The cancellation condition R|| x R |K ( x )| dx 0, R R >0 267 12 12 Then it can be shown that T is bounded on LRp  n  and satisfies a weak-type (1,1) estimate. Property is needed to ensure

K( x ) dx 0,  R12 , R  0 2 6 8 R12|| x R Which is quite easy to check. It is automatic, for instance, if K is an odd function. If, in addition, one assumes  and the following size condition

supR| x | 2 R |K ( x )| dx C , 2 6 9 R0

Then it can be shown that follows the smoothness condition . Is also often difficult to check in principle, the following sufficient condition of a kernel K can be used:

i. KCR 1(n {0}) ii. |Kx ( )| C ||x n1

Observe that these conditions are satisfied for the Hilbert and Riesz transforms, so this result is an extension of those result.

45 Singular integrals of non-convolution type: 18

These are even more general operators. However, since our assumptions are so weak, it is not necessarily the case that these operators are bounded on Lp .

2.6.3Definition:

A singular integral of non-convolution type is an operator T associated to a Calderón-Zygmund kernel K is an operator which is such that

gxTf    xdx   gxKx   yfydy   , 2 6 10

Whenever f and g are smooth and have disjoint support. Such operators need not be bounded on Lp .

Calderón-Zygmund operators:

2.6.4Definition:

A singular integral of non-convolution type T associated to a Calderón-Zygmund kernel K is called a Calderón-Zygmund operator when it is bounded on L2 , that is, there is a C > 0 such that

T f C f 2 6 11   L2 L2  

For all smooth compactly supported f .It can be proved that such operators are, in fact, also bounded on all with 1< p <.

2.6.1The T (b) theorem: The Tb  theorem provides sufficient conditions for a singular integral operator to be a Calderón-Zygmund operator that is for a singular integral operator associated to a Calderón-Zygmund kernel to be bounded on L2 . In order to state the result we must

46 first define some terms. A normalized bump is a smooth n  function  on R supported in a ball of radius 0,1 and centered at the origin such that   x 1, for all multi- indices  n 2. Denote by x   y    y x and n r  x  r x r  for all x in and r > 0. An operator is said to be weakly bounded if there is a constant C such that

x x n T r  y  r  y dy Cr . 2 6 12

For all normalized bumps  and . A function is said to be accretive if there is a constant C > 0 such that Reb x  c for all x in R. Denote byMb the operator given by multiplication by a functionb. The Tb  theorem states that a singular integral operator T associated to a Calderón-Zygmund kernel is bounded on L2 if it satisfies all of the following three conditions for some bounded accretive functions b1 andb2 :

(a)M TM Is weakly bounded; bb21

(b) Tb 1is in BM 0;

t t 18 (c)Tb 2  Is in BM 0, where T is the transpose operator ofT .

–  Hardy-Littlewood maximal function: 19 21

The usefulness of the Hardy-Littlewood maximal function M stems basically from two facts: 1) It is larger than the given function, since f Mf i.e., but it is 1 not too large, since Mfpp Cp f for 1 < p , while on L , M satisfies a weak type (1, 1) inequality.

47 2) It is more regular than the original function: If f is measurable, then Mf is lower semicontinuous. The fact that Mf controls f and its averages over balls (by definition) together with its Lp boundedness, leads to its frequent use in chains of inequalities, while its lower semicontinuity allows one to decompose its level sets using dyadic cubes. This is the basis of the often applied Calderon- Zygmund decomposition: Utilize Mf as a proxy for f , splitting the open set Mf tinto suitable disjoint cubes. This might be impossible to do directly with f t ,since in principle this set is merely measurable. Regarding derivatives, the study of the regularity properties of the Hardy-Littlewood maximal function is much more recent)

21 2.7.1Definition: Given a Lebesegue measureable function f on R, the maximal functions Mf   and Mf  are defined respectively by  1 u Mf supu x x f t dt: u  x   1 x Mf supx u u f t dt: x u 

u Mf  max MfMf  ,   sup ux 1 ftdtux  :     x 

Hardy-Littlewood maximal inequality: 19

2.7.2Definition:

This theorem of G. H. Hardy and J. E. Littlewood states that M is bounded as a sublinear operator from the space LRp  d  , p > 1 to itself. That is, if f Lp  Rd  , then the maximal function Mf is weak L1 bounded

48 and Mf Lp  Rd More precisely, for all dimensions d 1 and p d 1< p , and all f L R  , there is a constant Cd >0 such that for all > 0, we have the weak type-(1, 1) bound:

d Cd mxx R: Mf  < f . 2 7 1 d     LR1d     

This is the Hardy-Littlewood maximal inequality. With the Hardy-Littlewood maximal inequality in hand, the following strong-type estimate is an immediate consequence of the Marcinkiewicz interpolation theorem: there exists a constant

Apd, > 0 such that

Mf  A f   2 7 2 d d pd, d  d    LRLR       

Proof: For p , the inequality is trivial (since the average of a function is no larger than its essential supremum). For 1 < p  one proves the weak bound using the Vitali covering lemma.

Discussion: 19

It is still unknown what the smallest constants Apd, and Care in the above inequalities. However, a result of Elias Stein about spherical maximal functions can be used to show that, for1 1< we can remove the dependence of on the dimension, that is, AApd,  p for some constant Ap only depending on the value p . It is unknown whether there is a weak bound that is independent of dimension.

Application:

The Hardy-Littlewood maximal operator appears in many places but some of its most notable are in the proofs of the Lebesgue differentiation theorem and Fatous theorem and the theory of

49 singular integral operators the following statements are central to the utility of the Hardy- littlewood maximal operator. p a For f L Rn  , 1,p  Mf is finite i.e. b If ,then there exists a constant C such that, for all > 0: x Mf x   C f     Rn p c If , thenMf L Rn  and

M f p  Ap f . L Lp Where A depends only on p and c properties (b) is called a week type bound and csays the operator f M f  is bounded on LRp  n  . Property (b) can be proved using Vitali covering lemma. Property is clearly true when p , since we cannot take an average of p can then be deduced from these two facts by interpolation argument. It is worth noting does not hold for p 1. This can be easily proved by calculating M    where  is the characteristic function of the unit ball centered at the origin.

28–  Littlewood–Paley theory: 25

In harmonic analysis, Littlewood-Paley theory is a term used to describe a theoretical framework used to extend certain results about L functions to Lp functions for p  .It is typically used as a substitute for orthogonality arguments which only apply to functions when p . One implementation involves studying a function by decomposing it in terms of functions with localized frequencies, and using the Littlewood-Paley g-function to compare it with its Poisson integral. The 1-variable case was originated by J. E. Littlewood and Raymond Paley (1931, 1937, 1938) and developed further by Zygmund and Marcinkiewicz in the 1930s using complex function theory (Zygmund 2002,).E.M. Stein later extended the theory to higher dimensions using real variable techniques.

50 The dyadic decomposition of a function: 25

Littlewood-Paley theory uses a decomposition of a function f into a sum of functions fρ with localized frequencies. There are several ways to construct such decomposition; a typical method is as follows. If is a function on R, and ρ is a measurable set with characteristic function χρ , then is defined to be given by

ˆˆ ffρρ χ 2 8 1

Where the "hat" is used to represent the Fourier transform. Informally, is the piece of whose frequencies lie in .If  is a collection of measurable sets which (up to measure 0) are disjoint and have union the real line, then a well behaved function f can be written as a sum of functions forρ .When consists of the sets of the form

ρ [  2k11 ,  2 k ]  [2 k ,2 k ]. 2 8 2

For k an integer, this gives a so-called "dyadic decomposition" of ff: ρρ.There are many variations of this construction; for example, the characteristic function of a set used in the definition of can be replaced by a smoother function. A key estimate of Littlewood-Paley theory is the Littlewood-Paley theorem, which bounds the size of the functions in terms of the size of . There are many versions of this theorem corresponding to the different ways of decomposing f. A typical / p  estimate is to bound the L norm of  f by a multiple of ρρ the norm of .In higher dimensions it is possible to generalize this construction by replacing intervals with rectangles with sides parallel to the coordinate axes.

51 The Littlewood-Paley g function: 25 The g function is a non-linear operator on LRp  n that can be used to control the Lp norm of a function f in terms of its Poisson integral. The Poisson integral u x, yof is defined for y  0 by

u(,)()() x y P t f x t dt 283 Rn y   Where the Poisson kernel P is given by

2ππitx 2 | t | y ((ny 1) / 2) Py () xn e dt R π (nn 1)/2(|xy |22 ) ( 1)/2 2 8 4 2.8.1Definition:

The Littlewood-Paley g- function gf is defined by

 g( f )( x ) | u ( x , y ) |2 ydy 285 0  

A basic property of g is that it approximately preserves norms. More precisely, for p  , the ratio of the norms of and is bounded above and below by fixed positive constants depending on n and p but not on .

Applications:

One early application of Littlewood-Paley theory was the proof that if Sn are the partial sums of the Fourier series of a periodic p L function ( p > 1) and n j is a sequence satisfying nnj1 j >q for some fixed p > 1, then the sequence Sn converges almost j everywhere.

52 2– 9 Fefferman - Stien function and vector-valued 21 operators:

The sharp maximal function:

2.9.1Definition: For a locally integrable function f on Rn , the sharp maximal function f # is defined as regarding singular integrals. Suppose we have an operator T is bounded on LR2  n  so we have T f C f , 2 9 1   LL22    For all smooth compactly supported f .Suppose also that we can realize T as convolution against a kernel K in the sense that, whenever f and g are smooth and have disjoint support gxTf    xdx   gxKx   y f ydydx . 2 9 2 Finally we assume a size and smoothness condition on the kernel K then for r > 0, we have # r 1 T f  C f r  x. 293 For all xR n Let f be a measurable locally integrable function on , B an arbitrary ball in such that: f1 f x f dx, 2 9 4 BBB B   Then the maximal Fefferman-Stien function given by: 1 M# sup f x f dx 295 BXB B B   C.Fefferman and E.M.Stien prove the following extension of the Hardy- littlewood maximal theorem:

12 2.9.1Theorem : Let  fn n1 be a sequence of measurable function defined on .And let M be the well- known maximal operator given by 1 Mf x sup f y dy , xy 2 9 6 MQ  Q

53 Where the sup is taken over all cubes Q (or balls) centered at x and mx  is the Lebesgue measure of X . 1  r r 1 If 1rp   ,1    and if n1 fn   then we have

11 rr  rr   Mfnn   C  f    2 9 7 nn11 pp   Where C C r, p is independent of  fn  1 1  r r 1 d 2If 1,r   and ifn1 fn    L R; dm , then for every  0 we have:

11 d r rrC r  m x R:. Mf x    f    2 9 8 nn   nn11    1

Where C C r is independent of f and . 21    n n1

CHAPTER THREE

SEMIGROUPS OF OPERATORS

Our main question which we are going to answer in this chapter is what is the semigroup? It’s importance? It’s types? It’s properties? how can we generate it’s operators? What is it’s basic operators?

In mathematics, a semigroup is an algebraic structure consisting of a nonempty set S together with an associative binary operation. In other words, a semigroup is an associative magma. The terminology is derived from the anterior notion of a group. A semigroup differs from a group in that for each of its elements there may not exist an inverse; further, there may not exist an identity element. The binary operation of a semigroup is most often denoted multiplicatively: xy· , or simply xy , denotes the result of applying the semigroup operation to the ordered pair  xy,  .The formal study of semigroups began in the early 20th century. Semigroups are important in many areas of mathematics because they are the abstract algebraic underpinning of "memory less" systems: time-dependent systems that start form scratch at each iteration. In applied mathematics, semigroups are fundamental models for linear time-invariant systems. In partial differential equations, a semigroup is associated to any equation whose spatial evolution is independent of time. The theory of finite semigroups has been of particular importance in theoretical computer science since the 1950s because of the natural link between finite semigroups and finite automata. In probability theory, semigroups are associated with Markov processes.(Although the theory of semigroup of operators is usually considered as special topic in functional analysis , most of the well known special cases are

55 integral operators .Of these, are the Weierstrass and Poisson operators )

31  The semigroups : 26

3.1.1Definition:

A semigroup is a set S , together with a binary operation "·" that satisfies:

1. Closure: For all ab, in S , the result of the operation ab is also in . 2. Associativity: For all ab, and c inS , the equation a b  c  a  b  c holds.

More compactly, a semigroup is an associative magma.

Semigroup homomorphisms:

3.1.2Definition:

A homomorphism between two semigroups (,)S  and (S ,•) is a function f: S Ssuch that

x,:()()•() y  S f x  y  f x f y . 3 1 1

Any semigroup S may be embedded into a monoid (generally denoted asS1) simply by adjoining an element eS and defining e·· s s e sfor all s S{} e .

Examples of semigroups:

i. Semigroup with one element ii. Semigroup with two elements

56 iii. A monoid is a semigroup with an identity element iv. A group is a semigroup with an identity element and an inverse element. v. The set of positive integers with addition. vi. Square non-negative matrices with matrix multiplication. vii. Any ideal of a ring with the multiplication of the ring. viii. The set of all finite strings over a fixed alphabet Σ with concatenation of strings as the semigroup operation --- the so-called "free semigroup over Σ". With the empty string included, this semigroup becomes the free monoid over Σ. ix. A probability distribution F together with all convolution powers of F , with convolution as operation. This is called a convolution semigroup.

Special classes of semigroups: 26

i. A monoid is a semigroup with identity. ii. A subsemigroup is a subset of a semigroup that is closed under the semigroup operation. iii. A band is a semigroup the operation of which is idempotent. iv. Semilattices: A semigroup whose operation is idempotent and commutative is a semilattice. v. 0-simple semigroups. vi. Transformation semigroups: any finite semigroup S can be represented by transformations of a (state-) set Q of at most S 1states. Each element x of S then maps Q into itself x: Q Q and sequence xy is defined by qx y q xy for each q inQ . Sequencing clearly is an associative operation, here equivalent to function composition. This representation is basic for any automaton or finite state machine (FSM). vii. Bicyclic semigroups. viii. C0-semigroups. ix. Regular semigroups. x. Inverse semigroups.

57 Affine semigroup: a semigroup that is isomorphic to a finitely- generated subsemigroup of Z d . These semigroups have applications to commutative algebra

Structure of semigroups: 26 This section sets out concepts useful for understanding the structure of semigroups. Two semigroups S and T are said to be isomorphic if there is a bijection f: S T with the property that, for any elements ab, inS, f ab  f a f b . Isomorphic semigroups have the same structure.The semigroup operation induces an operation on the collection of its subsets: given subsets A and B of a semigroup, AB , written commonly as AB , is the setab a in A and b in B . In terms of this operations, a subset A is (i) a subsemigroup if AA is a subset of , (ii) a right ideal if AS is a subset of , and (iii) a left ideal if SA is a subset of A. If A is both a left ideal and a right ideal then it is called an ideal (or a two- sided ideal). An example of semigroup with no minimal ideal is the set of positive integers under addition. The minimal ideal of a commutative semigroup, when it exists, is a group.Green's relations are important tools for analyzing the ideals of a semigroup and related notions of structure. If S is a semigroup, then the intersection of any collection of subsemigroups of S is also a subsemigroup of S . So the subsemigroups of S form a complete lattice. For any subset A ofS there is a smallest subsemigroup T of S which contains A, and we say that AgeneratesT . A Single element x of S generates the subsemigroup { xn | n is a positive integer}. If this is finite, then is said to be of finite order, otherwise it is of infinite order. A semigroup is said to be periodic if all of its elements are of finite order. A semigroup generated by a single element is said to be monogenic (or cyclic). If a monogenic semigroup is infinite then it is isomorphic to the semigroup of positive integers with the operation of addition. If it is finite and nonempty, then it must contain at least one idempotent. It follows

58 that every nonempty periodic semigroup has at least one idempotent.

Semigroup methods in partial differential equations: 26 Semigroup theory can be used to study some problems in the field of partial differential equations. Roughly speaking, the semigroup approach is to regard a time-dependent partial differential equation as an ordinary differential equation on a function space. For example, consider the following initial boundary value problem for the heat equation on the spatial interval 0,1 R and timest 0:

u( t , x )  2 u ( t , x ), x  (0,1), t  0;  t x  u( t , x ) 0, x  {0,1}, t  0; 3 1 2  u( t , x ) u ( x ), x  (0,1), t  0.  0 Let X be the Lp space LR2 0,1 ,  and let A be the second- derivative operator with domain D( A ) { u  H2 ((0,1); R )| u (0)  u (1)  0}. 3 1 3

Then the above initial boundary value problem can be interpreted as an initial value problem for an ordinary differential equation on the space X :

 u( t ) Au ( t );  3 1 4 uu(0) 0 .

On a heuristic level, the solution to this problem "ought" to beu t exp tA u0 . However, for a rigorous treatment, a meaning must be given to the exponential oftA. As a function oft,exp tA , is a semigroup of operators from X to itself, taking the initial state u0 at time t 0 to the state u t exp tA u0 at timet . The

59 operator A is said to be the infinitesimal generator of the semigroup

32  Semigroup action: 27

3.2.1Formal definitions :

Let S be a semigroup. Then a (left) semigroup action (or act) of S is a set X together with an operation • : SXXwhich is compatible with the semigroup operation * as follows: i. for all st, in S and x in X,. sg t g x  s t g x 3 2 1

This is the analogue in semigroup theory of a (left) group action, and is equivalent to a semigroup homomorphism into the set of functions on X . Right semigroup actions are defined in a similar way using an operation • : XSX satisfying xg a g b x g a b.If M is a monoid, then a (left) monoid action (or act) of M is a (left) semigroup action of M with the additional property that ii. for all x in X: eg x x 3 2 2

Where e is the identity element of M this correspondly gives a monoid homomorphism. Right monoid actions are defined in a similar way. A monoid M with an action on a set is also called an operator monoid.A semigroup action of S on X can be made into monoid act by adjoining an identity to the semigroup and requiring that it acts as the identity transformation on X.

Terminology and notation:

If S is a semigroup or monoid, then a set X on which S acts as above (on the left, say) is also known as a (left) S-act, S-set, S- action, S-operand, or left act over S. Some authors do not

60 distinguish between semigroup and monoid actions, by regarding the identity axiom eg x xas empty when there is no identity element, or by using the term unitary S-act for an S-act with an identity. Furthermore, since a monoid is a semigroup, one can consider semigroup actions of monoids.The defining property of an act is analogous to the associativity of the the semigroup operation, and means that all parentheses can be omitted. It is common practice, especially in computer science, to omit the operations as well so that both the semigroup operation and the action are indicated by juxtaposition. In this way strings of letters from S act on X ,as in the expression stxfor st, in S and x in X.It is also quite common to work with right acts rather than left acts. However, every right S-act can be interpreted as a left act over the opposite monoid, which has the same elements as S, but where multiplication is defined by reversing the factors, sgg t t s,so the two notions are essentially equivalent. Here we primarily adopt the point of view of left acts.

Acts and transformations: 27

It is often convenient (for instance if there is more than one act under consideration) to use a letter, such as T,to denote the function TSXX:  323

Defining the S-action and hence write T s, xin place of sx.then for any sin S,we denote by

TXXs :  3 2 4

The transformation of X defined by

Ts  x T s,. x 325

61 Hence sT s is a rule assigning a transformation of X to each sin S . By the defining property of an S-act, T satisfies

TTTs t s t . 3 2 6

Conversely any such rule defines an S-act.

S-homeomorphisms: 27

Let X and Xbe S-acts. Then an S-homomorphism from X to Xis a map FXX:   3 2 7

Such that F sx sF xFor all sS and xS 3 2 8

The set of all such S-homeomorphisms is commonly written as HOMS  X, X  .M-homomorphisms of M-acts, for M a monoid, are defined in exactly the same way.

S-Act and M-Act:

For a fixed semigroup S , the left S-acts are the objects of a category, denoted S-Act, whose morphisms are the S- homeomorphisms. The corresponding category of right S-acts is sometimes denoted by Act-S (This is analogous to the categories R-Mod and Mod-R of left and right modules over a ring) .For a monoid M, the categories M-Act and Act-M are defined in the same way.

62 33 Transformation semigroup: 28

Important special classes of semigroup actions are the transformation semigroups. In this case, the semigroup S is a semigroup of transformations of a set X ,i.e., a collection of functions from X to itself that is closed under composition. Thus S is a subsemigroup of the monoid of all transformations of X.in this case the elements of the semigroup really are transformations of the set, so there is an obvious (tautological) action defined by evalutation: s·() x s x For s S,. x X 3 3 1

The characteristic feature of transformation semigroups, as actions, is st hat they are effective, i.e., if s·· x t x For all xX , 3 3 2

Then conversely if a semigroup S acts on a set X and the action is effective, then S is isomorphic to a transformation semigroup of X .

Any semigroup can be realized as a transformation semigroup via an analogue of Cayley's theorem: after adjoining an identity element if necessary, the action of the semigroup on itself by (say) left multiplication is effectiveation semigroups

3.3.1 Krohn–Rhodes theory:

Krohn–Rhodes theory, sometimes also called algebraic automata theory, gives powerful decomposition results for finite transformation semigroups by cascading simpler components.

63 34  Inverse semigroup: 29

In mathematics, an inverse semigroup S is a semigroup in which every element x in S has a unique inverse y in S in the sense that x xyx and y yxy . Inverse semigroups appear in a range of contexts; for example, they can be employed in the study of partial symmetries.

The basics:

The inverse of an element x of an inverse semigroup S is usually written x1. Inverses in an inverse semigroup have many of the same properties as inverses in a group, for example, 1 ab  b11 a . In an inverse monoid, xx1and xx1 are not (necessarily) equal to the identity, but they are both idempotent. An inverse monoid S in which xx111 x x , for all x in S (an unimportant inverse monoid), is, of course, a group.There are a number of equivalent characterisations of an inverse semigroup S :

1. Every element of S has a unique inverse, in the above sense. 2. Every element of S has at least one inverse ( S is a regular semigroup) and idempotents commute (that is, the idempotent of S form a semilattice). 3. Every L -class and every R -class contains precisely one idempotent, where L and R are two of Green's relations.

The idempotent in the L -class of s isss1 , whilst the idempotent in the R -class of s isss1 . There is therefore a simple characterization of Green's relations in an inverse semigroup: abLR aabb1   1, ab  aa  1  bb  1 3 4 1

64 Examples of inverse semigroups:

1. Every group is an inverse semigroup. 2. The bicyclic semigroup is inverse, with ab1  b11 a every semilattice is inverse. 3. The Brandt semigroup ( is a Rees matrix semigroup in which G is a group , AI   and is in BGIMGIAP ;;;;    and P is the identity matrix  PIGP ,0 if ij  which  ii ij is regular ) 56 is an inverse semigroup. 4. The Munn semigroup ( Let E be a semilattice . For each e in the set Ee i E: i e is a principal ideal of , the uniformly relation u on is given byu e,: f E E Ee; Ef . For each ef,  in we define T to be the set of all isomorphisms from Ee onto Ef . ef, LetT  T:, e f u we call T the Munn semigroup of E  ef    E the semilattice ) 57 is an inverse semigroup. Unless stated otherwise, ES  will denote the semilattice of idempotents of an inverse semigroup S .

The natural partial order: 29 An inverse semigroup S possesses a natural partial order relation  (sometimes denoted by ) which is defined by the following: a b  a  eb, 3 4 2

For some idempotent e inS . Equivalently, a b  a  bf , 343

65 For some (in general, different) idempotent f inS . In fact, ecan be taken to be aa1 and f to beaa1 .The natural partial order is compatible with both multiplication and inversion, that is, a b, c  d  ac  bd 3 4 4

And a b  a11  b . 345

In a group, this partial order simply reduces to equality, since the identity is the only idempotent. In a symmetric inverse semigroup, the partial order reduces to restriction of mappings, i.e.  if, and only if, the domain of  is contained in the domain of  and xx , for all x in the domain of .The natural partial order on an inverse semigroup interacts with Green's relations as follows: if st and s L t, then st . Similarly, if s R t on ES  , the natural partial order becomes: e f  e  ef , 3 4 6

So the product of any two idempotents in S is equal to the lesser of the two, with respect to ≤. If ES  forms a chain (i.e., ES  is totally ordered by ≤), then S is a union of groups.

Congruencies on inverse semigroups: 29

Congruencies are defined on inverse semigroups in exactly the same way as for any other semigroup: congruence  is an equivalence relation which is compatible with semigroup multiplication, i.e. a b, c  d ac  bd of particular interest is the relation , defined on an inverse semigroup S by ab There exists a cS with c a,. b 3 4 7

66 It can be shown that  is a congruence and that the factor semigroup S  is, in fact, a group. Indeed,  is the smallest congruence on S such that S  is a group, that is, if  is any other congruence on S with S  a group, then  is contained in . The congruence  is called the minimum group congruence onS .The minimum group congruence can be used to give a characterization of E-unitary inverse semigroups (see below).A congruence  on an inverse semigroup S is called idempotent pure if

a S, e  E ( S ), a e  a  E ( S ). 3 4 8

E-unitary inverse semigroups: 29

One class of inverse semigroups which has been studied extensively over the years is the class of E-unitary inverse semigroups: an inverse semigroup S (with semilattice E of idempotents) is E-unitary if, for all e in E and all S inS , es E  s  E. 349

Equivalently, se E  s  E. 3 4 10

35  Semigroups of operators: 30

We consider an abstract initial value problem u Au, u (0) u0in a Hilbert space H . The linear operator A is defined on some subset DA  of H . We want to identify a class of operators A for which we can, in some sense, define a ``solution"exp(At ) u0 . To this end, let us first review a few ways of defining the exponential of a matrix.

67 2 2 3 3 i. The power series:exp(At ) I  At A t  A t  . 3 5 1 26   ii. As a limit:exp(At ) lim ( I At )n . 3 5 2 n n   iii. By diagonalization or, more generally, transforming to Jordan canonical form. This is the procedure we followed in the preceding sections. iv. By Laplace transforms: Consider the equation

u Au, u (0) u0 , 3 5 3

And take the Laplace transform:  st uˆ()(). s 0 e u t dt 3 5 4

We find the transformed equation suˆˆ u0 Au, 3 5 5 1 Which leads to uˆ (). sI A u0 3 5 6 The inversion formula for the Laplace transform then yields u()(), t1  i est sI A1 u ds 3 5 7 2i  i 0   Where σ must be chosen larger than the real part of any eigenvalue of A. This leads us to the definition exp(At )1  i est ( sI A )1 ds . 358 2i  i   For unbounded operators, the power series definition is not useful. For A d22 dx . Then formally, we would have

nn2 1 nn t d u exp(At ) u t A u 2n . 3 5 9 nn00nn!!dx For the right hand side to make any sense u must have derivatives of arbitrarily high orders, and convergence poses even more serious restrictions. In addition, we would have to impose an

68 infinite number of boundary conditions onu . The definition as a limit seems to suffer from the same defect. However, we can make the following modification to it: exp(At ) lim ( I At )n . 3 5 10 n n  

The difference is now that we are dealing with powers of an inverse operator rather than powers of A. It turns out that this definition is indeed useful. Indeed, the following theorem, known as the Hille-Yosida theorem, is at the foundation of the study of infinite-dimensional evolution problems.

36  Hille–Yosida theorem: 30 34

In functional analysis, the Hille–Yosida theorem characterizes the generators of one-parameter semigroups of linear operators on Banach spaces. The theorem is mainly of theoretical interest, the Lumer-Phillips theorem is more useful in determining whether a given operator generates a strongly continuous semigroup. The theorem is named after the mathematicians Einar Hille and Kosaku Yosida who independently discovered the result around 1948.

3.6.1 Statement of the theorem:

Assume that A is a linear operator defined on a dense subspace DA  of a Hilbert space H . Assume further that there are constants

69 1 M and such that  AIσ  exists as an operator from H to DA  for σω and

().AI n M 3 6 1 () n  

Then exp(At ) u lim ( IAt )n uexists, moreover, exp(At )is a n n bounded operator from H to itself. We have the exponential propertyexp(At )exp( As ) exp( A ( t s )), and the continuity property lim exp(At ) u u. Ifu D() A , then t0 d exp(At ) u A exp( At ) u . 3 6 2 dt   The set of operators exp(At )where t 0is referred to as a semigroup of operators; it is closed under multiplication since.exp(At ) exp( As )  exp( A ( t  s )) .It is in general not possible to extend t to negative values; for instance the heat equation is well posed only for solutions forward in time, not for solutions backward in time. We note that

(IAIAt )n  (  1) n (  n )  n ( t )  n . 3 6 3 n t n   The condition guarantees that, fornt , the norm of the right hand side is bounded by

M ()t n n  M . 3 6 4 (nt )nn (1 ) tn

Forn , the right hand side converges toMtexp( ).

70 30 34 One-parameter semigroup of operators:

3.6.1 Formal definition:

If X is a Banach space, a one-parameter semigroup of operators on X is a family of operators indexed on the non-negative real numbers {Tt ( )}t [0, ) such that

1. TI(0)  3 6 4 2. T( s t )  T ( s ) T ( t ),  t , s  0. 3 6 5

The semigroup is said to be strongly continuous, also called a (C0) semigroup, if and only if the mapping t T() t x 3 6 6

 Is continuous for all xX , where 0,  has the usual topology and X has the norm topology. The infinitesimal generator of a one- parameter semigroup T is an operator A defined on a possibly proper subspace of X as follows:

1. The domain of A is the set of x X such that h1(()) T h x x 3 6 7 Has a limit as h approaches 0 from the right.

2. The value of Ax is the value of the above limit. In other words Ax is the right-derivative at 0 of the function t T(). t x

The infinitesimal generator of a strongly continuous one-parameter semigroup is a closed linear operator defined on a dense linear subspace of X .The Hille–Yosida theorem provides a necessary and sufficient condition for a closed linear operator Aon a Banach

71 space to be the infinitesimal generator of a strongly continuous one-parameter semigroup.

Hille-Yosida theorem for contraction semigroups: 30 35

In the general case the Hille-Yosida theorem is mainly of theoretical importance since the estimates on the powers of the resolvent operator that appear in the statement of the theorem can usually not be checked in concrete examples. In the special case of contraction semigroups (M 1 and  0in the above theorem) only the case n 1has to be checked and the theorem also becomes of some practical importance. (See Dr .Adam‟s Ph.d as application of semigroup in a total perceived power differential games theory from functional analysis view point) 58

31 37  C0-semigroup:

In mathematics, a C0-semigroup, also known as a (strongly continuous) one-parameter semigroup, is a homomorphism from  R,into a topological monoid, usually Lb  , the algebra of linear continuous operators on some Banach spaceB , that is continuous in the topology. Thus, strictly speaking, not the C0- semigroup, but rather its image, is a semigroup. A C0-semigroup is a type of one-parameter semigroup, a generalization of a one- parameter group.

Example

C0-semigroups occur for example in the context of initial value problems of the form, dx f( x ); x (0) x , Cauchy‟s problem CP 3 7 1 dt 0    

72 Where x and f take values in a Banach spaceB .If the solution of CP is unique (depending on f ) for x0 in some given domain D B , one has the "solution operator" defined by

(t ) x0 x ( t ),Where xt  is the solution of CP . Thus one can view  as an "evolution operator", and it is clear that one should have ()()()s  t  s  t 3 7 2

On the domain D this is just the condition of a semigroup- morphism. Then one can study the conditions under which  is continuous for the topology on Lb  induced by the norm on , which amounts to check that

lim (t ) x  x  0 For each x in D 3 7 3 t0 00 0  

3.7.1Formal definition: 31

All that follows concerns the following definition:

A (strongly continuous) C0-semigroup on a Banach space B is a map

:()RL B 3 7 4

Such that

i. 0 I :  id (identity operator on )   B ii.  t,s  0 :  t  s   t  s iii. x0 B :  t x 0  x 0  0 , as t 0

73 The first two axioms are algebraic, and state that it is a map of semigroups; the last is topological, and states that the map is

continuous in the strong operator topology.

The Infinitesimal generator: 31

3.7.2Definition:

The infinitesimal generator A of a C0-semigroup  is defined by

A xlim1 (  ( t )  I ) x 3 7 5 t0 t Whenever the limit exists the domain of ADA,  , is the set of xB for which this limit does exist.t  May also be denoted by the symbol

te tA . 3 7 6

This notation is compatible with the notation for matrix exponentials, and for functions of an operator defined via the spectral theorem.

The Stability:

The growth bound of a semigroup  (on a Banach space) is the constant

 lim1 log (t ) 3 7 7 t0 t It is so called as this number is also the infimum of all real numbers  such that there exist a constant  M 1 with

74 ()t Met For all t 0 3 7 8

The semigroup is exponentially stable, i.e. K, a  0,  t  0:  ( t )  K eat 3 7 9

If and only if its growth bound is negative. One has the following: 3.7.1 Theorem: A semigroup is exponentially stable if and only if for every xB there is C 0 such that

2 t  dt C. 3 7 10 R  

Analytic semigroup: 32

In mathematics, an analytic semigroup is particular kind of strongly continuous semigroup. Analytic semigroups are used in the solution of partial differential equations; compared to strongly continuous semigroups, analytic semigroups provide better regularity of solutions to initial value problems, better results concerning perturbations of the infinitesimal generator, and a relationship between the type of the semigroup and the spectrum of the infinitesimal generator. (See Dr Adam‟s Ph.d as application of semigroup infinitesimal generator in a total perceived power differential games theory from functional analysis view point) 58

3.7.3Definition :

Let t exp At be a strongly continuous one-parameter semigroup on a Banach space  X ,  with infinitesimal generator A. is said to be an analytic semigroup if

75 i. For some 0 < θ < π ⁄ 2, the continuous linear operator exp At : X X can be extended to

tεθ , θ {0}  {tt £ :| arg( ) | θ },

And the usual semigroup conditions hold for s, t θ : s , tε θ

exp A0   id , exp A t s exp At exp As ,And, for each xXε , exp At x is continuous in t;

ii. And, for all tε θ 0, exp Atis analytic in t in the sense of the uniform operator topology

Characterization: 32

The infinitesimal generators of analytic semigroups have the following characterization: A closed, densely-defined linear operator Aon a Banach space X is the generator of an analytic semigroup if and only if there exists an ωε Rsuch that the half- plane Reλω  is contained in the resolvent set of A and, moreover, there is a constant C such that

C RA   3 7 11 λ λω

For Re (λ) > ω If this is the case, then the resolvent set actually contains a sector of the form

π λεC : arg λ ω   δ 3 7 12 2

For someδ  0, and an analogous resolvent estimate holds in this sector. Moreover, the semigroup is represented by

1 1 exp At  eλt λλ id A d , 3 7 13 2πi γ

76 Where γ is any curve from eiθ  to such that γ lies entirely in the sector

λεC : arg λ ω θ 3 7 14

Withπ22 θ  π  δ

Symmetric diffusion semigroup: 33

3.7.4Definition:

A symmetric diffusion semigroup is a collection of linear operators p Ttt0 defined on Ld,   over a measure space ,d  satisfying the following properties

T0  Id, TTTts  t s, Tt LLpp 1, p 1, ;

2 2 limTft  In L , fL ; 3 7 15 t0

38 Special integral operators: 24

Weierstrass Operators:

3.8.1Definition:

The Weierstrass operators Waa,  0,is defined by

Waa f  W f , 3 8 1 1  n 2 2 Where Wa  x 4π  exp x 4 a,it is clear that

Wa  x  Ua  W x, 3 8 2

77 Where 1 1 1 1   n 2 W x 4π  2 exp x 4 a ,anda 2 a2 ,2 a 2 ,....,2 a 2 , a  0     Also, 1  n W t dtπ 2 exp  t2 exp  t 2 ....exp  t 2 dt RRnn   12    n  n 1  π 2 exp t2 dt  1 3 8 3 R    In the expression, the integral, which is usually evaluated by the methods of a complex variable, is obtained from the n-dimensional case 1 1 1  nq 2 q nq11  W4π a 2 exp t q 4 a dt  ak2 , a q Rn    q 3 8 4

Where kq is finite forq 1, it follows from Young „s inequality that if rp ,then 1 nq11  W f a2 k f 1q 1  1 r  1 p . 3 8 5 aq  r p,        

The Weierstrass operators Wa, has various properties which connect n 22 it with the Laplacian operator x j . j1

3.8.1 Theorem :( the „semigroup property‟ of Weierstrass operator) 24

The class Waa,  0, Weierstrass operators is such that i. For 1p   , W f  f ,and a   pp  W f f 0 Anda  0 , f Lp Rn , a   p  ε  

78 Pn ii. Fora0, b  0, Wa W b f  W a b  f  ,  fε L R, p 1, n 1 . Proof: To prove (i) we have from 3.8.1Definition 24 The Weierstrass operators Waa,  0,is defined by

Waa f  W f , 1  n 2 2 Where Wa  x 4π  exp x 4 a, it is clear that

Wa  x  Ua  W x, Where 1 1 1 1   n 2 W x 4π  2 exp x 4 a ,anda 2 a2 ,2 a 2 ,...,2 a 2 , a  0.     Also, 1  n W t dtπ 2 exp  t2 exp  t 2 ....exp  t 2 dt RRnn   12    n  n 1  π 2 exp t2 dt  1 R    In the expression, the integral, which is usually evaluated by the methods of a complex variable, is obtained from the n-dimensional case 1 1 1  nq 2 q nq11  W4π a 2 exp t q 4 a dt  ak2 , a q Rn    q 3 8 4

Where kq is finite forq 1, it follows from Young „s inequality that if rp ,then

79 1 nq11  W f a2 k f 1q 1  1 r  1 p .And from aq  r p,       n 24 p  Theorem Let fL R , where np1,1   , let  n 1 hL R and let    n h t dt . Then  R n (i) Uhf  h f p a R . a   p 1   (ii) U h f f  0 ( as a  0 ) a   p To prove (ii) we have from Lemma: Let 1  n 2 Wa 4π a 2 exp x 4 a Then for , WWW ab0, 0 a b a b And from h f g  f  h  g . It follows that

Waa Wb W  W b  f  W a b  f  W a b  f 

The Poisson operator and its conjugate:

The Poisson operator and its conjugate arose primary from the theory of harmonic functions. They also have connection with the Hilbert operator in one-dimension and Riesz in n-dimension n  2 .Further results gives connections between the operators and the Fourier transform. The function ua and uak, , a 0, k 1,2,..., n are defined by 1 1 2 n 1 2 n 1 u x c a a2 x 2 ; u x c x a2 x 2 , an    a, k  n k   3 8 6

80 1  n1 1 2 Where the constant cn is defined to be n 1 π .On making 2 2 the change of variable b a2 t  4ξ it follows that 1  3 n 2 b2 exp a2 t 4 b db 0    

1 1 n 1 3 n 2    2 a2  t42 b exp  1 b db    0 1 n 1 11n 2    22 a2  t42 b exp  b db ,    0 So that 1 1 3 n 12n 1    2 a u t 4π  2 b2 exp  a  t 4 b db . 3 8 7 a 0     It follows applying Fubini‟s theorem, that u x dx 1. Rn a  

The operator Pa , a 0, Pak, , k  1,2,..., n are defined by

Pa  f x  ua f x; Pak,  f   uak,  f x. 3 8 8 Pa Is usually called the Poisson operator, and in the n-dimensional case n  2,the operator, Pak, , k 1,2,..., nare the n components of the conjugate operator Pa it follows by applying Fubini‟s theorem to the integral representation given above for ua ,that 1 1   3 2 ab2 4 a P f 4,π 2 b e W f db 3 8 9 a     0 b     Where Wb the Weierstrass operator is defines in the previous section. n Sinceua  Ua  u,wherea  a, a ,..., aε R and 1 n 1 2 2 u x  cn 1; t 

81 By change of the variables yj x j t j , j 1,2,..., n in each of the one dimensional integrals xj , j 1,2,..., n it‟s clear that 1 1 q n1 q 1 2 nq1 n 1 q 1 u a1,  t2 dt  k a aqq Rn    3 8 10 1 1 q n1 q 1 2 nq1 n 1 q 1 u a1,  t2 dt  a k ak, q Rn   q  3 8 11

The constant kq being finite ifq 1, and kq being finite ifq 1.Hence it follows from Young‟s inequality, that p nnr i. If rp , then PLRLRa :     , and n11 r p P f a k f , 1q 1  1 r  1 p . 3 8 12 aq  r p        And p nnr ii. If rp , then kn1,2,..., , PLRLRak, :      and n11 r p P f  a k f , 1q 1  1 r   1 p .3 8 13 ak, r q p  

The classPaa ,0  satisfies semigroup properties similar to those for Weierstrass operator. We consider these in the following theorem

3.8.2 Theorem: 24

The classes of operatorsPaa ,0 andPak, , a 0, k 1,2,..., nare such that, for fε LPn R,1 p   , a  0, b  0,

82 i. P f f,0 P f  f  as a  0 aa  ppp  

ii. Pa P b f  P a b  f , iii. Pb,,, k P a f  P a P b k f P a b k  f , n iv. Pb,, k P a k f  P a b  f , k1 Proof 24

Ornstein–Uhlenbeck operator: 34

In mathematics, the Ornstein–Uhlenbeck operator can be thought of as a generalization of the Laplace operator to an infinite- dimensional setting. The Ornstein–Uhlenbeck operator plays a significant role in the Malliavin calculus.

The finite-dimensional picture:

The Laplacian:

Consider the gradient operator  acting on scalar functions f:, Rn  R the gradient of a scalar function is a vector field  f : Rnn  R . The divergence operator div, acting on vector fields to produce scalar fields, is the adjoint operator to . The Laplace operator  is then the composition of the divergence and gradient operators:  div  , 3 8 14

Acting on scalar functions to produce scalar functions. Note that A  is a positive operator, whereas  is an operator. Using spectral theory, one can define a square root 1  1/2for the operator 1   .This square root satisfies the following relation

83 involving the Sobolev H1-norm and L2-norm for suitable scalar functions f :

2 2 1 1 2 f f H1    . 3 8 15 L2

The Ornstein–Uhlenbeck operator: 34

Often, when working on Rn, one works with respect to Lebesgue measure, which has many nice properties. However, remember that the aim is to work in infinite-dimensional spaces, and it is a fact that there is no infinite-dimensional Lebesgue measure. Instead, if one is studying some separable Banach space what does make sense is a notion of Gaussian measure; in particular, the abstract Wiener space construction makes sense. To get some intuition about what can be expected in the infinite-dimensional setting, consider standard Gaussian measure  n on Rn : for Borel subsets A n/2 2 of  n A : 2π e x p x 2 dx . 3 8 16   A      This makes  RBRn,, n  n  into a probability space; E will denoted expectation with respect to .The gradient operator  acts on a (differentiable) function  :RRn  to give a vector field  :RRn .The divergence operator  (to be more precise, n,since it depends on the dimension) is now defined to be the adjoint of  in the Hilbert space sense, in the Hilbert space LRBRR2  n,,; n  n  in other words,  acts on a vector field  :RRnn to give a scalar function  :RRnn and satisfies the formula EE[f · v ] [ f v ]. 3 8 17

84 On the left, the product is the point wise Euclidean dot product of two vector fields; on the right, it is just the point wise multiplication of two functions. Using integration by parts, one can check that  acts on a vector field v with components vi, i  1 , ..., n , as follows:

n vi v()()(). x x vi x x 3 8 18  i   i1xi

The change of notation from “div” to “δ ” is for two reasons: first,  is the notation used in infinite dimensions (the Malliavin calculus); secondly is really the negative of the usual divergence. The (finite-dimensional) Ornstein–Uhlenbeck operator L (or, to be more precise, Lm ) is defined by L :  , 3 8 19

With the useful formula that for any f and g smooth enough for all the terms to make sense, ()·.f g   f  g  fLg 3 8 20

The Ornstein–Uhlenbeck operator L is related to the usual Laplacian  by Lf( x )  f ( x )  xg  f ( x ). 3 8 21

The Ornstein–Uhlenbeck operator for a separable Banach space: 34

Consider now an abstract Wiener space E with Hilbert space H and Wiener measure .let D denote the Malliavin derivative. The Malliavin derivative is an unbounded operator from LER2  ,  ;  into LEH2  ,  ;  in some sense; it measures “how random” a function on E is. The domain of is not the whole of

85 LER2  ,  ;  but is a dense linear subspace, the Watanabe-Sobolev space, often denoted by D1,2 (once differentiable in the sense of Malliavin, with derivative in L2 ).Again,  is defined to be the adjoint of the gradient operator (in this case, the Malliavin derivative is playing the role of the gradient operator). The operator  is also known the Skorokhod integral, which is an anticipating stochastic integral; it is this set-up that gives rise to the slogan “stochastic integrals are divergences” Satisfies the identity

EE[,][]DF v H  F v 3 8 22

For all F in D1,2 and u in the domain of .Then the Ornstein– Uhlenbeck operator for E is the operator Ldefined by L :. D 3 8 23

CHAPTER Four

CHAPTER FOUR

ESTIMATION OF VECTOR –VALUED SINGULAR

INTEGRALS WITH SEMIGROUPS OF OPERATORS

Our main questions which we are going to answer here is How can we develop generalized Littlewood-Paley theory for semigroups acting on Lp -spaces of functions with values in smooth or convex Banach spaces? and how this general theory becomes more satisfactory (and easier to be handled) in virtue of the theory of vector-valued Calder´on-Zygmund singular integral operators? To develop a generalized Littlewood-Paley theory for semigroups acting on -spaces of functions with values in uniformly convex or smooth Banach spaces. It is well known that martingale inequalities involving square function are closely related to the corresponding inequalities concerning the Littlewood-Paley or Lusin square function in harmonic analysis. It is in this spirit that a 49 generalized Littlewood-Paley theory is developed in [XU] for functions with values in uniformly convex Banach spaces. The main goal is to extend the results in [XU] to general symmetric diffusion semigroups, and thus to develop a generalized Littlewood-Paley theory for these semigroups on Lp -spaces of functions with values in uniformly convex or smooth Banach spaces. We characterize, in the vector-valued setting, the validity of the one-sided inequalities concerning the generalized Littlewood-Paley-Stein g -function associated with a subordinated Poisson symmetric diffusion semigroup by the martingale cotype and type properties of the underlying Banach space. Recall that a symmetric diffusion semigroup is a collection of linear operators p Ttt0 defined on Ld,   over a measure space,d . 4.2.1theorem, states that a Banach space  is of martingale

87 cotype q if and 0nly if for every symmetric diffusion semigroup

Ttt0 with subordinated semigroup Pt t0 the validity of the reverse inequality (with a necessary additional term) characterizes the martingale type q (see 4.2.2Theorem). These results are proved in section 42  .The main ingredient of our arguments is the classical Rota theorem on the dilation of a positive contraction on Lp by conditional expectations. This theorem allows to reduce 4 1 14 (after discretization) to a corresponding inequality for martingales. This approach via Rota’s theorem is also efficacious in studying and its dual form for an individual semigroup. We shall show in section 43  that for a given subordinated Poisson semigroupPt , is equivalent to its dual form, which is an inequality reverse to with , p and q replaced byB , p and q , respectively (and with an additional term). The key to this is the existence of a certain projection, whose proof, using Rota’s theorem once more. Our proof for the implication “ martingale cotype q uses the Poisson semigroup (Note however that this Poisson semigroup on the torus is a multiplicative semigroup on(0, ) ). Thus it would be interesting to know the family of semigroups for which the validity of implies martingale cotypeq . We shall show in section 44  and 45 in the case of the usual Poisson semigroup and the Poisson semigroup subordinated to the Ornstein-Uhlenbeck semigroup on Rn , this general theory becomes more satisfactory (and easier to be handled) in virtue of the theory of vector-valued Calder´on-Zygmund singular integral operators. For these operators, under suitable conditions, one can get the equivalence of the strong type (,)ppand the boundedness BMO− (see4.4.1 theorem. As a consequence, we obtain the characterization of the Lusin co-type in terms of

88 boundedness of the g-functions 4.4.2Corollary and 4.5.2 , 4.5.3theorems .The two sections 44 and 49 45 extend most of the results in [XU] for T to Rn .The previous results for the usual Poisson semigroup on can be extended to the Poisson semigroup subordinated to the Ornstein- Uhlenbeck semigroup on . This is done in section 46  (see 4.6.1 , 4.6.2theorems).The last section47  contains a further characterization of Lusin cotype property in terms of almost sure finiteness of the generalized Littlewood-Paley g -functions (4.7.1 , 4.7.4theorems).

41 Developing a generalized Littlewood-Paley theory for  semigroups acting on Lp -spaces:

It is well known that martingale inequalities involving square function are closely related to the corresponding inequalities concerning the Littlewood-Paley or Lusin square function in harmonic analysis. It is in this spirit that a generalized Littlewood- Paley theory is developed in[XU] for functions with values in uniformly convex Banach spaces. The main goal is to extend the results in [XU] to general symmetric diffusion semigroups, and thus to develop a generalized Littlewood-Paley theory for these semigroups on Lp -spaces of functions with values in uniformly convex or smooth Banach spaces.

Martigale cotype and type of Banach space : Given a martingale  fn with values in a Banach space , its generalized “square” function is defined as: 1  q q Sqn  f fn1 4 1 1 n1 B

89 Then  is said to have martingale cotype qq,2 , if there exist p 1, and a constant C 0 such that S f Csup f 4 1 2 qLp n n Lp   B p For every bounded -valued L -martingale fn. The validity of the reverse inequality defines martingale type q ,12q . Recall that if the inequality above (or its inverse) holds for one p 1, , so does it for all p 1, .

 Lusin co-type and type of Banach space : Let f be a function in L1T  , where T denotes the torus equipped with normalized Haar measured . The classical Littlewood-Paley g -function is defined for zT as: 53 1 1 2 2 dr 2 Gf z 01  r  Pr  f z  413 1r In this notation, 1 2 2 2 Pr 1 pr Pr  f t   f t   f t  4 1 4 r r   With 1r2 Pr    415 1rr2 2 cos

Being the Poisson kernel for the disk. It is a classical result that for any p 1, , there exist positive constantscp and Cp such that c f fˆ 0.  Gf  C f 4 1 6 ppLLLppTTT     p     One can extend the definition of G to functions that take values in a Banach space , just by replacing absolute value by norm in . In this case, holds if and only if  is

90 isomorphic to a Hilbert space. However, one of the two inequalities in 4 1 6 can be true in non Hilbertian spaces. The study of these one-sided inequalities is the main objective of [XU] 49 . More generally, we can introduce the following generalized “Littlewood-Paley g -function” 1q 1 q q dr Gqr f z 01  r  P  f z  B 4 1 7 1r Then  is said to be of Lusin cotype q (resp. Lusin typeq ) if there exist p 1,  and a positive constant C such that:  G f C f resp.0 f C fˆ G f q Lp T Lp T Lp T   q Lp T   B   B   B   4 1 8 It is not difficult to see that if  is of Lusin cotype q (resp. Lusin type q ), then2q  (resp.12q ). It is proved in[XU] that the definition above is independent of p , that is, if one of the inequalities above holds for one p 1, , then so does it for every p 1,  (with a different constant depending on p ). The main result of [XU] states that a Banach space  is of Lusin type (resp. Lusin cotype q ) if and only if  is of martingale type q (resp. martingale cotype q ).

Extending the previous results to general symmetric diffusion  semigroups:

The main goal is to extend the results in [XU] to general symmetric diffusion semigroups, and thus to develop a generalized Littlewood-Paley theory for these semigroups on Lp -spaces of functions with values in uniformly convex or smooth Banach spaces. Recall that a symmetric diffusion semigroup is a collection

91 p of linear operators Ttt0 defined on Ld,   over a measure space,d  satisfying the following properties T Id, TTT T 1, p 1, ; 419 0 ts t s, t LLpp   2 2 limTft  In L , fL ; 4 1 10 t0  2 TTtt On L , if f  0, Tt11 4 1 11

The subordinated Poisson semigroup Pt t0is defined as u tu2 4 1 e t e Pt f00T fdu3 Tu fdu. 4 1 12 u tu2 4 2 u 2 is again a symmetric diffusion semigroup, see[ST1] 47 .

Recall that if A denotes the infinitesimal generator ofTtt0, then 1 that of is A 2. It is well known (and easy to check) that p  any bounded operator T on L for all p 1,  naturally and p boundedly extends to LB  for every Banach space, where denotes the usual Bochner-Lebesgue Lp -space of -valued functions defined on  .More precisely, the extension isT Id . Indeed, this is clear for p 1 (via projective tensor product); the case p  is done by duality, and then 1p by interpolation. With a slight abuse of notation (which will not cause any ambiguity), we shall denote these extensions still by the same symbol T .Thus Tt and Pt have straight forward extensions to p LB  for every Banach space ; moreover, these extensions are also contractive. (Note that we can also justify these extensions by the positivity of Tt and ).According to the convention above, we shall consider Ttt0 and as semigroups on too. In these circumstances we can define the generalized “Littlewood- Paley g -function” associated to the semigroup as:

1 q q  P f dt  f x   t t 4 1 13 q   0 t t B

The first result of this work is 4.2.1 theorem, states that a Banach space  is of martingale cotype q if only if for every symmetric diffusion semigroup Ttt0 with subordinated semigroup Pt t0 p the generalized g -function operator q is bounded from LB  to Lp  , namely p . q  f p C f Lp  , fL B   4 1 14 L  B   The validity of the reverse inequality (with a necessary additional term) characterizes the martingale type q (see 4.2.2theorem).

The n -dimensional generalized “Littlewood-Paley g-function:

For anyq 1, the -dimensional generalized “Littlewood-Paley g- function” is defined as 52 1  q q q Gq  f x 0 t  Pt  f x 2 , 4 1 15 B Where 1 2 2 2 n PPtt Pt  f x 2   f x   f x , 4 1 16 B txB k1 k B With n1  2 t Pxt    n1 n1 , 4 1 17  2 2 2 2 xt 

93 The kernel of the Poisson semigroup for the upper half space. Note that we use the same symbol Pt to denote the Poisson kernels both on T and on Rn . This should not have any confusion.

42 One-sided vector-valued Littlewood-Paley-Stein  inequalities for semigroups:

We shall consider general symmetric diffusion semigroups, that is, p the collections of linear operators Ttt0 defined on L  , satisfying419 and 4 1 11 .Given such a semigroup

Ttt0we consider its subordinated semigroupPt t0 , defined as in4 1 12. LetF L2   be the subspace of the fix points of , i.e., the subspace of all f such that Pt  ff  for allt 0. Let FL: 2  F be the orthogonal projection. It is clear that F extends to a contractive projection (still denoted byF) on for every 1p and that FL p  is exactly the fix point space of on .Moreover, for any Banach space , F extends p to a contractive projection on LB  for every 1p and that p FL is again the fix point space of considered as a B semigroup on .According to our convention, in the sequel, we shall use the same symbol F to denote any of these contractive projections.Recall4 1 13 that the generalized Littlewood-Paley g-function associated with is defined by 1 q q  P fx  f x  t t . 4 2 1 q   0 t t B The main results of this section are the following two theorems.

94  4.2.1 Theorem Given a Banach space  and2q , the following statements are equivalent: (i) is of martingale cotype q .

(ii) For every symmetric diffusion semigroupTtt0with subordinated semigroupPt t0and for every (or, equivalently, for some) p 1, there is a constantC 0 such that p

q fp C f p , fL B  . 422 L  LB 

. Theorem Given a Banach space  and12q , the following statements are equivalent: (i) is of martingale type .

(ii) For every symmetric diffusion semigroup Ttt0with subordinated semigroup and for every (or, equivalently, for some) p 1, there is a constantC 0 such that  f C F f   f , 4 2 3 Lp    Lp  q Lp    B   B    

The rest of this section is essentially devoted to the proof of these theorems. The difficult part is the implication “(i) (ii)” in theorem. Then the same implication in theorem will follow by duality. Both converse implications will be done by 49 using the Poisson semigroup on the torus with the help of [XU] . For the main part of the proof we shall need the following result, which has independent interest.

4.2.3Theorem Let  be a Banach space of martingale cotype  q 2, and Ttt0a symmetric diffusion semigroup. Then for any p 1, ,

95 1 q q  Mft dt p t C f p , fL , 424 0 t t pq,,B L  B   B B Lp  Where M1 t T ds. t t 0 s The key ingredient is Rota’s dilation theorem (see 4.2.5Theorem below), which allows to reduce the inequality in 4.2.3Theorem to a similar inequality for martingales. Given a  -finite measure spaceM,,F dm and a sub-σ- algebraGF , we denote as usual by E G the conditional expectation with respect to G. Note that our measure space is no longer a probability one; however all usual properties on conditional expectations in the probabilistic case are still valid in the present setting. Recall that is a positive p  contraction on L M,,F dmfor every p 1,  and naturally p extends to LB  M,,F dmfor every Banach space .The classical Doob maximal inequality is also valid in the vector-valued setting. LetFn be an increasing filtration of sub -algebras ofF . For p f LB  M,,F dm we define its maximal function as: # f sup Ef Fk  4 2 5 n1 B Then we have the following Doob maximal weak type 1,1 inequality # λλm f# f x dm x 4 2 6   f λ   B      For every Banach space  .Similarly, we can also extend the results [MT] 42 ; in particular, we get that for every 1,pq  and p every sequence fk   LB  M,,F dm

96 q 1q 1 # q q f C f . 4 2 7  kk pq, B   kk11  Lp Lp We shall use the following lemma, motivated by [ST1] 47 .

 4.2.4Lemma Let  be a Banach space of martingale co-type

 q 2, , M , dm be any  -finite measure space and En be an arbitrary monotone sequence of conditional expectations on M, dm .Then, for every pp,1 ,

1q  q1 q  nn n1 f C p , q ,B f Lp 4 2 8 n1 B B Lp Where EE   σ  0 n 4 2 9 n n 1

Proof:

If we define dnn E En1 forn0 (with the convention that E ), we have 1 0 2j 2 j1 2 j  E  E  dk   d k   d k . j 22jj1 kk00k21j1 Consider, for each nJ, the unique integer such that n JJ1 22nnn . Then

97 k 1 Jn 2 n  nn1  jdjj   jd nn1 k0 kJ1  J 2 k2 n1 Now, for eachk,0 k Jn , k 1 k 1 k k 2 2 2 kk2 k jd22    j d  2    d j kk11jjk   k jj2 2 1 k 1 j21ij 2k 1  2k k  E jk   i21k 1

We can treat the rest of the terms in a similar way, and then we get 1  Jn 2k 1  k nn1  2 k E i  k  nn1 k0k 1   i21 n1  nE   E   . n  Jnn11 k J  kJn 1  Thus q 1 q  q1 J  q n n k nq1 f 2   n n1  q q k n0 B nk10nn1   B q 1 q q1 J k 1  n n 2    E  q q ik  nk10nn 1 i21k 1 B

1 q  nq1 q  nE  q q nJ n 1  n1 nn 1 B

98 q 1 q nq1 n 1 E  q q kJ 1  J n n1 nn 1 k21n B I  II  III  IV q n i nq 1 n i q Using i122ai  i12 ai , we have that  JJ q 1 Jq 1 nnkkqq1 I 2n  2 ff   2  q1 kkBB2 n1n k  0n k  0  q   k f k0 B Since Bis of martingale cotype I p Cf. L pq,,B Lp B In order to handle the second term, let us call k jk when 2kk1 j  Then, q J 1 Jq 1 n n q 1122 Jq 1 II q E  f 2 n E  f qq11jj  kj   nn11nnjj00kj  B B  1 n1 q 1 q EfEf  2  j  kj   j kj  n1 n j0 B J 1 B Using an4-2-3 d the martingale cotype q ofB, we obtain that 1 q  1 q II p Ef L  j  kj   k0 j1 B Lp

1 q k  q 2 1 C  f C f . pq, k B pq,,B Lp j0 j21k 1 j1 B Lp

Analogously, one can show that III IV C f LLLpppq,,B q B We shall also need the following result due to Rota  Assumption of Rota theorem:

Let Q be a linear operator on Lp ,, A d  satisfying the conditions  i. Q LLpp 1 for every p 1, , ii. QQ  in L2, iii. Qf 0, for every f  0, iv. Q1 1.

4.2.5 Theorem For any Q as above, there exist a measure spaceM,,F dm, a decreasing collection of  -algebras

...FFFFFnn1  ...  1  0  ,and another -algebra f Fˆ  F such that: a) There exists an isomorphism i:,,,,A d  MFˆ dm which induces an isomorphism between Lp -spaces, also denoted byi , i( f )( m ) f ( i1 m ), 4 2 10 b) For every fε Lp  M,,F dm , we have 21n  ˆ Q i f x  E En  f i x x 4 2 11 Where Eˆ f E f Fˆ And E f E f F . 4 2 12     nn     

Proof of 4.2.3theorem:

100  Let  be a Banach space of martingale cotype q 2, and

Ttt0a symmetric diffusion semigroup. Then for any p 1, , q 1 Mf q  tt dt  C f , fL p , 0 t t pq,,B Lp  B   B B Lp  Where M1 t T ds. t t 0 s Observe that it is enough to prove

1 q q  Mft dt t C f p , 4 2 13 0 t t pq,,B L  B B Lp  for any 0ab   and also that it is enough if we restrict ourselves to functions f in the algebraic tensor productB Lp  . K 47 Take then fv k1 kk . By the results in [ST1] , see the lemma in and its proof, it is not difficult to observe that for everyt0 0,, there exists 0  0 such that:  j Tt f x  fj  x t t0  4 2 14 j0 j For t t0  0, t 0  0  and almost every x , where  f j p 0  LB and where f j depend ont0 . Since we can cover (,)ab with a finite collection of such intervals, we can split into a finite collection of subintervals (,)abii of in which an expression like holds for a fixed t0 (and therefore, with the same f j ) for every tε  aii, b .Then, splitting the integral between a and binto the integrals corresponding to such subintervals, we can handle all the functions appearing in the integral as power series with vector valued coefficients. In these circumstances, we can replace the integral by Riemann sums, and all derivatives by

101 difference quotients. The first step is choosing  small. Then, we approximate the integral as follows: 11 qq qq n  b M fdt 1 q1 M f  tntt      , a ttt  nn tn   BB  0  LLpp 4 2 15 Where the sign means that the difference term goes to zero as  0 . The next step is substituting the partial derivative inside the sum by the difference quotient M f M f n1 n n1   1 1  T fds 1 1 n T fds 4 2 16  n1 0 ss n  0 And then each of the integrals by its Riemann sums, getting then that

1 1 q q q q n1 nn1 b Mft dt q1 11 a t n  Tjj f  T f tntnnn 1 jj00 B p 0 B L Lp

n 1q 1 q1 q n f f  n n1 B nn 0 Lp 4 2 17 Where f 1 n T f . Now, observe that by our hypothesis, n n1 j0 j T 2satisfies assumptions (i)-(iv) of Rota’s theorem and ˆ ˆ Tn f E En  f . Hence, nnf E f  where n is as in 4.2.4lemma therefore, by the properties of conditional expectation and Lemma, we get

102 q 1q 1 n1 q q b Mft dt q1 ˆˆ  t n En f  E f a t t  nn  n1  B  B 0 Lp Lp 1 1 1q q  q1 q nnn f1 f   nn fn1 f n1 B p p L L q B

Cfpq,,B Lp 4 2 18 B

Therefore, we have achieved the proof of 4.2.3theorem The following lemma says that the boundedness of p p Gq f Tf q from L T in L T is equivalent to the L0,1, dr 1 r B     B     boundedness of the operator T when the kernel is restricted to values of r close to one and  close to zero.

 4.2.6Lemma Let be a Banach space and pq,  1, .Let

 0 (close to 0).Then there is a constant C (depending only on ) such that for any fL p T B   Pr 1r  0,1  r  f  C f Lp T . 4 2 19 r   Lp T B   dr   Lq 0,1 , B   1r And

Pr 1rχ r χ θ  f  C f p .   1δ ,1    δ , δ    δ L T  r Lp T B q dr   L 0,1 , B   1r 4 2 20

103 Proof:

The proof is very easy. We show only the first inequality. Its left hand side is a convolution of the -valued function f with an q dr L 0,1 , -valued function. Therefore it is enough to prove 1 r that the latter is in L1 , namely, we have to prove qdr L 0,1 , 1r 2π P γ  1rr χγ r d  . 4 2 21 0   0,1δ      r qdr L 0,1 , 1r But this follows immediately if we observe that q P γ  1 rrr χ   0,1δ   r   qdr L 0,1 1r q 2 22 1δ 2 1rr  2  1 sinγ 2 dr 1 rC  q .    22δ 0 1rr 2 sin2 γ 2  1 r 4 2 22 Hence the lemma is proved The following easy lemma is proved in a similar way as in [ST1] 47

 4.2.7 Lemma Let  be a Banach space and pq,  1, .Then for any fL p  B we have qq11 P f qq  M f  tttdt  C  t dt  , 4 2 23 00tttt 0   BB   

Where C0 is an absolute constant.

104 Proof:

1 e14s Call  s  Using integration by parts we have 2  s32

1 s   s  s   Pt 20 2 sMss ds    0 4   2  M ds   0 s   s M2 ds t t s t  t  ts 4 2 24

M s Therefore withM s  s

  tP  2 ts2 2 sMds    2 sstsMds   2  . 4 2 25 t t 00t22 s t s

Thus

q 1 1 q q q Pt f dt   2 dt t 2 s s  t sM f ds 0t tt  0  0 ts2 B B 

1q 11q  q dt  2 K tM f 4 2 26 0 t B   t Where  K 0 s s ds 4 2 27 Hence the lemma is proved. Now we are well prepared for the proofs of 4.2.1 and 4.2.2 theorems .  Proof of theorem: Given a Banach space  and2q , the following statements are equivalent: (i) is of martingale cotype q .

105 (ii) For every symmetric diffusion semigroupTtt0with subordinated semigroupPt t0and for every (or, equivalently, for some) p 1, there is a constantC 0 such that p

q fp C f p , fL B  . L  LB 

(i)  (ii) this is an immediate consequence of4.2.3 Theorem and 4.2.7Lemma 1 (ii) (i). We shall prove that the operator f Gq  f  is bounded p p from LB T  to L T  for p 1,  recall that q 1q 1 q P dr G1 f z 1,  rr  f z  zT 4 2 28 q   0  rr1 B 49 By , this is equivalent to the martingale cotype q of . Observe that if in the Poisson kernel Prr,0 1 we change the parameter t , according tore we obtain the kernel Pt of the Poisson semigroup subordinated to the heat semigroup in the torus. Fix a  0,1 (very close to 1). By the same change of parameter and the 1et fact that for any t0, log , t , then we have et q q 1 P drlog 1 ettP e dt 1r r  f t  f    r1  r0 et  t 1et q P Cq log tt f  dt  ,q 0 t   t q P Cq  tt f  dt . 4 2 29  ,q 0 t   t   Therefore, by hypothesis (ii), we have that

106 P 1.rr  r  f  C f 4 2 30    ,1    ,q Lp T   r   p B   L  Lq0,1, dr T     B 1r Then by 4.2.6Lemma we get 1  Gq  f p  C f Lp T 4 2 31 L T  B   49 By [XU] , this implies that is of Lusin cotype q ,and so of martingale cotype too. Thus the proof of 4.2.1 theorem finished.

 Proof of 4.2.2 theorem: Given a Banach space  and 12q , the following statements are equivalent: (i) is of martingale type q .

(ii) For every symmetric diffusion semigroup Ttt0with subordinated semigroupPt t0and for every (or, equivalently, for some) p 1, there is a constantC 0 such that p f C F f   f , fL . Lp    Lp  q Lp  B   B   B    

(i) (ii) .Write the spectral decomposition of the semigroup

Pt t0: For any fL 2  pPf  et de f , 4 2 32 t 0   

Wheree is a resolution of the identity. Thus P f t   et de f . 4 2 33 t 0   

107 It is easy to deduce from this formula that for f, g L2  (recalling that F is the projection on the fix point subspace of Pt t0   PPttf g dt  f F f g  F g d  4  0  t t d ttt 4 2 34 p p Now we use duality. Fix two functions fL B  and gL B  where pdenotes the conjugate index of p . Without loss of generality we may assume that f and g are in the algebraic tensor p 2 p 2   products LL  B and LL  B          Respectively. With , denoting the duality between  andB, we have fgd,,,  FfFgd       fFfgFgd          4 2 35 The first term on the right is easy to be estimated:

FfFgd,   Ff Fg Ff g p        LLpp  Lp    L   BB  B    B 4 2 36 For the second one, by and Holder’s inequality

 PPttfgdt f F f,, g  F g d   0 t t d ttt

 PPttfgdt  0 t t d  ttt

 q  fg p    .4 2 37 L  q Lp Now since  is of martingale typeq , B is of martingale co- typeq. Thus by 4.2.1theorem,

g C g p 4 2 38 q    L    LP  B 

108 Combining the preceding inequalities, we get  f, g d  F f  C q f g p 4 2 39    LLp   p L   B     B Which gives (ii), taking the supremum over all g as above such that g Lp  1. B  (ii)(i). As in the corresponding proof of 4.2.1theorem, we use again the Poisson semigroup on the torus. We keep the notations introduced there. Recall that PPt  et . By the calculations done there, q q P 11rPr  fdr  log tt  f dt , 4 2 40   r  1  r  t   t   Where the equivalence constants depend only on  andq . On the other hand, on the interval (0,δ ),we have q q ttP  1rPr  fdr  1 et  f e dt 0  r  1  r log eett  t   1 q qt1 P Cq  e  t f dt  ,q log t   q q P Cqq t1 t f dt 4 2 41   ,q  log t     Therefore, q P q tqq1 t  f dt  C1 1.  rPr  f dr 4 2 42 00t   ,q    r  1  r   Thus by hypothesis (ii) ˆ 1 fp  C f0. G f 4 2 43 L T  ,q   q   Lp T   B   B   49 Hence by [XU] , is of Lusin type q , and so of martingale type too.

109  43 Duality:

Throughout this section Ttt0 will be a fixed symmetric p diffusion semigroup defined on Ld,   , and Pt t0 its subordinated Poisson semigroup. We shall keep all notations introduced in the previous section for these semigroups. In particular F is the contractive projection from Lp  (also p from LB  onto the fix point subspace of . The following is the main result of this section.

4.3.1Theorem Let  be a Banach space and1,pq .Then the following statements are equivalent: (i) There is a constant C 0 such that: f C f , fL p 4 3 1 q Lp  Lp  B       B   (ii)There is a constant such that: p gp  C F g p   g , gL   432 L  L  q p B B B L  The proof of the implication (i)(ii) of 4.2.2theorem shows in fact (i)(ii) in the theorem above. The inverse implication needs much more effort (as usually in such a situation). Let q dt p A  LRB , .An element h in LB  is a function of two t variables x andtR  , i.e. ,h:,. x t ht  x The key to the implication (ii)(i) above is the existence of a bounded projection p from LA  onto the subspace of all functions which can be P f written as h x  tt  x for some function f on  . Formally, t t Pt Qh the desired projection is given by h t x where Qh is t defined by

110 P h Qh x   ttt x dt , x 4 3 3 0 t t   p Note that Qh  is well-defined for nice functionshL A  , for instance, for all compactly supported continuous functions from p R to LB  .By the density of all such functions in ,to prove the boundedness of Q we need only to estimate the relative norm of Qh for all such h  4.3.2Theorem LetB,,pq, be as in 4.3.1 theorem. Then for any (nice)

qQh p C p, q h Lp . 434 L  A Consequently, qQextends to a bounded operator from to Lp  with norm controlled by a constant depending only on p and q. Admitting this theorem, we can easily prove Theorem

Proof of Theorem (Let  be a Banach space and1,pq .Then the following statements are equivalent: (i) There is a constant C 0 such that: f C f , fL p q Lp  Lp  B     B   (ii)There is a constant such that: p gp  C F g p   g , gL   L  L  q p B B B L 

(i)(ii). The proof for this is similar to that for (i)(ii) in 4.2.2Theorem p p (ii)(i). Fix an fL  . Choose hL  of unit norm B      LRq dt , B t such that

111 Pt f dt fp t x,. h x dx 4 3 5 q L   R t t t Now we apply 4.3.2theorem toB, pand q′. (We may assume f and h are nice enough to legitimate the calculations below). We have, by hypothesis (ii) and Theorem P h f f x,, ttt x dt dx q Lp  R  t t

  f x, Q h x dx

 fp Q h L    Lp  B   B  C f  Qh  C f 4 3 6 LLP q   p p   B   L  B   This yields (i). As for 4.2.3 theorem, we shall reduce theorem to an analogous inequality for martingales via Rota’s theorem. Let Enbe a monotone sequence of conditional expectations as in 4.2.4Lemma Let us maintain the notations in that lemma and its proof. In the remainder of this section lq denotes the usual q 1 space over N but with weight , i.e., the norm of a sequence a n is given by 1 q 1 q aalq   n 4 3 7 n1 n q q The corresponding -valued version islB , denoted by D lB in the sequel. Now we consider the discrete version of Q defined p by4 3 3. As before, the elements in LMD   are regarded as p p sequences with values in LMB   . Given h LD  M  we define 1 Rh nnn h  , 4 3 8 n1 n

112 EE... Where      . Recall that  0 n . Rh Is clearly nnn1 n n1 well-defined for finite sequences h .  n n

4.3.3 Lemma LetB,,pq be as in 4.3.1 theorem. Let Enbe an arbitrary monotone sequence of conditional expectations on a measure space M, dm .Then for any finite p sequenceh  hn  LD  M 

nn Rh C p, q h p . 4 3 9   n1 LMp LM  D   D Consequently,h n Rh extends to a bounded operator  n  n1 p on LMD   with norm majorized by a constant depending only on p andq .

 Proof

44  Poisson semigroup on Rn :

This section and the next are devoted to the study of the Littlewood-Paley g-function on Rn in the vector-valued case. Our main goal is to prove the implications (ii)(i) in 4.2.1 and 4.2.2 theorems in the particular case of the Poisson semigroup on . This section collects some results on the g-function operator on , represented as a Calder´on-Zygmund operator. It can be considered as preparatory, although some of these results are of general interest. The proof of the mentioned implications will be done in the next section. Let  be a Banach space and 1p .Recall the generalized Littlewood-Paley g-function on :

113 1q  q q dt n Gq  f x 0 t  Pt  f x 2 , xR 4 4 1 B t It is often easier to consider the corresponding g -function defined only by the derivative in time, which is the following 1 P q q G1 f x  tq t f x dt 442 q   t t B 2 Similarly, we defineGq  f  as the part ofGq  f  corresponding to the gradient in the space variable: 1 q q G 2 f x tq  P  f x dt , 4 4 3 q    0 x t   2   B t Where  ,..., 444 x    xx1 n These g -functions can be treated as Calderón-Zygmund operators. To this end we first recall briefly the definition of these operators. Given a pair of Banach spaces B1 andB2 , a linear operator T is a Calderón-Zygmund operator on Rn , with associated Calderón-Zygmund kernel k if T maps LR  n  the c,B1 space of the essentially bounded functions on Rn with compact support, into -valued strongly measurable functions on , and for any function f L  Rn  we have : c,B1 Tf x   k x,, y f y dy For a. e. x outside the support of f , T

Where the kernel, k x,, yL BB12 satisfies: (a) k x, y  C x y n .  n 1 (b) xyk x,, y   k x y  C x  y

We shall always assume that there is εL BB12, such that

T c  c for all cB1 .

114 Let us recall the BMO and H1 spaces on Rn . Let  be a Banach n space BMOB  R is the space of -valued functions f defined on such that: 1 fsup f x  f dx   . 4 4 5 BMO Rn Q B  Q Q B 1 Where f  f x dx and the supremum is taken over the Q Q Q cubesQR n. The space H1 is defined in the atomic sense.  n Namely, we say that a function a LB  R  is a B -atom if there exists a cube QR n containing the support of a, and such 1 that aQ n  , and a x dx 0. Then, we say that a function LRB   Q 1 n f is in HRB   if it admits decomposition fa iii, whereai are -valued atoms and i i . we defined f 1 infi i  , HB where the infimum runs over all those decompositions 35 . n The following theorem is a kind of folklore. BMOc,B  R  n Denotes the subspace of BMOB  R  consisting of functions with compact support.

 4.4.1 Theorem Let BB12, be two Banach spaces and T a Calder´on-Zygmund operator with an associated kernel k as above. Let S be defined as S f  T f  . Then, the following B2 statements are equivalent:  n n (i) maps LRc,B   into BMOB  R  . 1 2 (ii) maps into BMO Rn . 1 n 1 n (iii) maps HRB   into LRB   . 1 2

115 (iv) T maps LRp  n  into LRp  n  for any (or equivalently, for B1 B2 some)1p . (v) maps LR1  n  into LR1,  n  . B1 B2 (vi) maps BMO Rn  into BMO Rn  . c,B1 B2 (vii) S maps into BMO Rn . Proof: The structure of the proof is the following: first, (i)(ii) (iii) (i). Then, we prove (i) (iv) (vi) (vii) (ii) and (iv) (v) (i). The fact that L is contained in BMO gives that (vii) implies B1 B1

(ii). Since the norm of a function in BMOB is a function in BMO, then we have (i) ( ii) and (vi) (vii). To get (ii) (iii) and (iii) (i), we can proceed as in 40 with minor modifications due to considering the operator Sf Tf . B2 As we already know that (i) (iii), we can apply interpolation 36 and we have that T maps Lp into Lp for1p , so we have B1 B 2 (i) (iv). The proof of (iv) (vi) is where the condition that T c x  c plays a role. Let f be a function in BMO . Given a cubeQ with c,B1 center x0 , let Q be its doubled cube. We decompose 1  Tf x Tg20 x  dx  Q Q B2 11Tg x dx  Tg x  Tg x dx, 4 4 6 QQ1  B 2  2 0    Q 2 Q B2 Where f g g , g f f  and g f  f  f . By 121  Q  Q 2  QQ RQn using Jensen and the Lp boundedness ofT , we have 11pp pp    1 1   1  QQRTg1 x dx  Tg 1 x dx  n g 1 x dx Q QQ  B1     

116 1 p p 1 Q f x  fQ dx  Cp f 4 4 7 Q B BMOB 1 1 Where in the last inequality we have used the John-Nirenberg theorem. On the other hand, usingT c x  c , we have     TgxTgxTff   00   xTff      x   22 QQ RQRQnn     kxy,,.  kxy fy  f dy RQn     0    Q  Now, using the hypothesis on the kernel k we have that

n xx 0 for x Q, y R Q , k x,, y  k x0 y  n1 and therefore yx 0  TgxTgx 2.J 1 fyfdy  22   0  B  22JJQQ1 n   Q 2 J1 yx 0 B1   j 1  2.j f y  f dy  C f 4 4 8 j 2 Q Q B BMO j1 2 Q 1 B1

By [GR] 38 , the strong type  pp,  implies that T is of weak type( , ).this gives (iv)  (v). From (v), statement (i) can be achieved by using an slight modification of the argument given in the proof of lemma [GR] .The key is using Kolmogorov’s inequality (In probability theory, Kolmogorov's inequality is a so-called "maximal inequality" that gives a bound on the probability that the partial sums of a finite collection of independent random variables kn  1 , ..., exceed some specified bound) 55 relating L1, norm q with L norm for 01q and the fact that BMOq  BMO.It is well known that the various Littlewood-Paley g -functions can be expressed as Calder´on-Zygmund operators with regular vector- 49 valued kernels (see [XU] for the case of the torus; also see [ST2] 48 for the scalar case). Therefore, we immediately get the following.

117  4.4.2Corollary Given a Banach spaceB,q  1,, the following statements are equivalent.  n n (i) Gq maps LRc,B   into BMO R . 1 n 1 n (ii) Gq maps HRB   into LR  . p n p n (iii) maps LRB   into LR for any (equivalently for some) p 1,  , n n (iv) maps BMOc,B  R  into BMO R . 1 n 1, n (v) maps LRB   into LR  . 1 These statements are also equivalent if we replace by Gq or 2 Gq . The following result reduces the boundedness on , and to that on one of them.

4.4.3 Proposition Let  be a Banach space and pq,  1, .

p n Then for any f LB  R 

12 GGqq ff    , 4 4 9 LRLRpp nn   Where the equivalence constants depend only on pq, andn .

(For proof see)

Remark. The equivalence in proposition still holds when Lp is replaced by L1, . The last result of this section is a duality theorem for the boundedness of the g -functions (4.4.5 theorem below). This theorem is a particular case of 4.3.1 theorem.It is also the 49 analogue for Rn (for the torus). As in such a situation the key

118 is again the existence of a certain projection. Fix a Banach space p dt andq 1, , and with A  LRB  ,. the projection in question is t defined as (recalling thattttP) Q h x  h., t x dt , xR n. 4 4 10   0 t t   Note that Qh  is well defined for functions h in a dense family p n of LRA   , for instance, for those which are compactly supported n continuous functions on RR  .The proof of the following lemma is an adaptation for Rn .

 4.4.4Lemma Let  be a Banach space and pq,  1, . Let Q be defined as before. Then for any -valued continuous function h with compact support in G1 Q h C h 4 4 11 q   p n p, q LRp n   LR  A  1 Consequently, the map hGq  Q h extends to a bounded map p n p n from LRA   to LR .

Proof:

f Q h , , 4 4 12 Then s f  s     h., t x dt ss0 t t st 2 P  h., t x dt  k  h ., t x dt , 4 4 13 00s ttt s, t   Where 2 k st 2 P  st P 4 4 14 s, t s t u2 u u s t Now consider the operator Kx :AA defined by K x s   k x  t  dt 4 4 15 0 st, t  

119 For every A Using the inequality st kst,  x  C 4 4 16  x s t  n2

And a similar one for the derivative of kxst,   in x , one can easily check that for any x Rn 0, K x is bounded and C C Kx   n , Kx  , 4 4 17 x x n1 Thus  x, y K x yis a Calder´on-Zygmund kernel. Hence to prove the lemma it suffices to show that the singular integral p n operatorh K h is bounded on LRA   in virtue of 4.4.1 Theorem. This is easily done as follows. For xR n and s 0, we have 1  dt q K h x, s 0 Rn kst,  x  y dy t 1  q dt q 0 Rn kst,  x y h y, t dy t 1  q dt q C0 Rn kst,  x y h y,. t dy 4 4 18 t Therefore, q q  q dt ds KhLRq n  CRRnn 00   kxyhytdyst,     ,  dx A  ts  q st dsq dt Cnn dx h y, t dy RR 00   n2 st x y  s  t     q q Chqn, LRq n . 4 4 19 A  This implies the desired boundedness of the singular integral on q n LRA  .

120  2 Remark. 4.4.4lemma holds as well for Gq and Gq instead 1 ofGq . Moreover, the weak type , inequality is true too. From lemma and using the arguments in the proof of 4.3.1 theorem, we deduce the following

4.4.5 Theorem Let  be a Banach space andq 1, . Then the following statements are equivalent: (i) One of the statements in 4.4.2corollary holds. (ii) For every p 1,  (equivalently for some ) there is a constant C such that 1 q n fCp  G , f L R LRn q p n B   B  LR 

45  Continuation of Poisson semigroup on Rn :

Our aim in this section is proving that in the definition of the Lusin type or cotype the -function on the torus can be replaced by that 49 on Rn . This, together with , will imply the validity of (ii)  (i) in both 4.2.1 and 4.2.2 theorems for the particular case of the Poisson semigroup on . This is done by a careful analysis of the Poison kernels on T and on R and a comparison of its essential parts. We shall also need a lemma, similar to 4.2.6lemma, for the Poisson kernel on .

4.5.1 Lemma Let B,,pq and  be as in the previous lemma. p n Then for any f LB  R  P tt  t x f C f . 4 5 1 0,  ,    LRp   t     B   LRp q dt   LB 0, , t

121 Now we state our result on the Lusin cotype for the Poisson semigroup on Rn .

 4.5.2 Theorem Let be a Banach space and2q . Then the following statements are equivalent: (i) is of Lusin co-typeq . (ii) For every (or equivalently, for some) positive integer n and for every (or equivalently, for some) p 1, there is a constant C 0 p n Gq  f p n  C f LRp n , f LB  R  4 5 2 LR  B   (iii) For every (or equivalently, for some) positive integer n there is a constant C 0 such that

Gq  f 1, n  C f LR1 n , 4 5 3 LR  B   1 2 The same equivalences hold with Gq or Gq instead ofGq in (ii) and (iii).

Proof: In virtue of 4.4.3 Proposition, we need only to prove the theorem for . (i)  (ii). This is a particular case of 4.2.1 theorem. (ii)  (iii).This equivalence for a given integer is already contained in4.4.2 corollary. (ii) for n > 1 (ii) for n = 1. By corollary, it is enough to  get the boundedness from Lc,B into BMO of on R from the same boundedness property of on . n1  To this end, consider x  x2,....., xn  R , andh Lc,B  R,and, n define f x  h x1   n1  x where x  x12, x ,....., xn  R . 0,1  The symmetric diffusion semigroup generated by the Laplacian on is given by convolution with the Gaussian density. Then we have

122 xy 2 1  T f x e4t f y dy t   Rn n 2   4t 2 xy  11 1 4t  1 C0R e h y dy 1 C 0 Tt h x 1, 4 5 4 4t 12 1 1 1 Where Tt is the heat kernel in R. If we denote by Pt the Poisson 1 semigroup subordinated to on and by Pt the Poisson kernel on , the formula 4 1 12 implies that 11 Pfxt  PP t fxC  0 t hx 1  CPhx 0 t   1 , 4 5 5 And therefore 11 GGqqf x  C01 h x . 456 Now, for every interval IR consider QI n the cube in Rn whose sides are the interval I . Then, 111 1C0 1 QIIGGGqfxdx  n n C q hxdx  ..... dx n  q hxdx  . Q I 0 1 1I 1 1 457 Therefore, and also by using similar arguments, C 1 GGGG1f x  1 f dx 0 1 h x  1 h dx 4 5 8 Q QIq qQ I q 11  q I Hence, GG11h1 f  C f  C h . 4 5 9 qqBMO R BMO Rn LRLR n      C0   B   B   ii) For n = 1  (i). By 4.4.2 corollary and the corresponding 49 q result in for the torus, we need to prove that for every fL B T  1 Gq  f q  C f Lq T . 4 5 10 L T  B   1 (Note that we take pq here; also recall that Gq is the Gq -function on the torus relative to the derivative in the radius). But by 4.2.6 lemma it is enough to show that for  0 very close to 1

123 Pr   1r 1 ,1 r   ,      f  C f Lq T r     q B   L T  q dr L 0,1 B  1r 4 5 11 By the change of variablesre t , we have tt2 22 P   2 1ee  4 1   sin 2 r rk  . r 1, 2 t tt2 2 1ee 4 sin 2  4 5 12 Thus is reduced to q q t k f   d dt d  Cq f 4 5 13 T 0  t     q T   t LB   Where  log 1 . It is elementary to decompose k   as follows 1 t k k0   k 1   k 2  , 4 5 14 tt    tt      Where t22 kt0   2      4 5 15 t 2 0,    ,     t22  1 2 And wherekt   and kt   are supported on 0,  ,   and satisfy kC1   t , kC2    , 4 5 16 t  t22 t  The verification of this decomposition, though entirely elementary, could be tedious. One way to do this is to replace each time only one term of et and sinQ 2 by their respective equivalents 1t and  2 in kt   and in the functions so successively obtained. At each stage the difference between the old and new functions is of type when is replaced or of type when is replaced. It is evident that

124 q 2 dt q q T 0 t  k f   d  d  C f q . 4 5 17 t t  LB T  1 It is also easy to get such an inequality for kt φ indeed, we have q q  dtqq   t t k1  f   d  d  C t f d  T 00 t        22Lq T t t  B   q q Cfq, Lq T 4 5 18 B   Therefore, 4 5 13 is reduced to q 0 dt q q T 0 t  k f   d  d  C f q . 4 5 19 t t LB T  0 Now we use the Poisson kernel Pt on R. By the definition of kt in 4 5 15 , Px  k0  x  1 t  t  x. 4 5 20 t 2 t 0,    ,    

Put f x  f x  ,   x for xR . Then by4.5.1 Lemma, we see that is further reduced to q q  P dt q  tt  f x dx C f . 4 5 21 R 0 t t LRq B   This last inequality follows from hypothesis (iii). Therefore, is of Lusin co-typeq , and thus the theorem is proved. The following is the dual version of 4.5.2 theorem.  4.5.3Theorem Let  be a Banach space and 12q Then the following statements are equivalent: (i)  is of Lusin type . (ii) For every (or equivalently, for some) n 1 and for every (or equivalently, for some) p 1, there is a constant C 0 p n fLRp n  CGq  f  p n , f LB  R . 4 5 22 B   LR  1 2 The same equivalence holds withGq or Gq instead of Gq in (ii). Proof:

125

(i)  (ii) is a particular case of 4.2.2 theorem (ii) (i) is done by duality in virtue of 4.4.5and 4.5.3theorems .

 46 Ornstein-Uhlenbeck semigroup:

Our purpose of this section is to extend the results in the previous one to the Poisson semigroup subordinated to the Ornstein- Uhlenbeck semigroup on Rn . Recall that this latter semigroup is defined by et x y O f x1 exp f y dy . 4 6 1 t   nt22Rn      2t 1e  1e    O We denote by t t0 the Poisson semigroup subordinated to Ott0 as defined in 4 1 12 .

4.6.1Definition: Let  be a Banach space and1q . As for the usual Poisson semigroup on , we introduce the Littlewood- Paley g -function associated to :

1q  q q dt n gq  f x 0 tOt f x 2 , xR 4 6 2 B t n Here  still denotes the gradient in RR  . We shall also consider its two variants corresponding to the time derivative and the space variable gradient, respectively: q 1q O f g1 f x   tq t  x dt , 4 6 3 q   0 t t   B And 1q 2  q q dt gqx f x 0 tOt f x 2 . 4 6 4 B t

126 The following is the analogue of 4.5.2theorem for the Ornstein- n Uhlenbeck semigroup. n Stands for the Gaussian measure on R , 2 i.e.n expx dx . 

 4.6.1Theorem Let be a Banach space and2q . Then the following statements are equivalent: (i)  is of Lusin co typeq . (ii) For every (or equivalently, for some) positive integer n and for every (or equivalently, for some) p 1, there is a constantC 0 p n gq  f p n  C f LRp n, , f LB  R ,.n  4 6 5 LR ,n  B  n  (iii) For every (or equivalently, for some) positive integern there is a constantC 0 such that 1 n gq  f 1, n  C f LR1 n, , f LB  R ,.n  466 LR ,n  B  n  1 2 The same equivalences hold with gq or gq instead of gq in ii) and iii).We also have a similar result for Lusin type.

4.6.2Theorem let  be a Banach space and 1q 2. then the following statements are equivalent: i)  is of Lusin type . ii) For every (or equivalently, for some) n 1and for every (or equivalently, for some) there is a constantC 0

p n f C fd g f , f L R ,. LRp n, Rn nqB   p n B  n  B  n  LR ,n  467 The same equivalence holds with or instead of in (ii).The proofs of the theorems above can be reduced to those on the usual Poisson semigroup on already considered in the previous section. The usual technique dealing with operators

127 related to the Ornstein-Uhlenbeck semigroup consists in decomposing Rn into two regions: one where the Gaussian and Lebesgue’s measure are equivalent, and the corresponding operators comparable, and the other where the kernels of the operators can be estimated by a well behaved positive kernel. This technique was invented by Muckenhoupt in the one-dimensional case, and extended by Sjögren to higher dimensions, for the maximal operator. For vector valued functions, the technique has been developed in [HTEV] 39 . Following this, for the g -function n operator, define the domains in RR  : nn3 23nn D x,: y x  y    And D x,:. y x  y    1   1xy 2   1xy  4 6 8

Let  be a smooth function on which is supported on D1 , equal to 1 on D2 and satisfies 1 xy x,,. y   x y  C x  y 4 6 9 Let T be a Calder´on-Zygmund singular integral operator on with kernel k x, y as described at the beginning of section 44 (and satisfying the conditions (a) and (b) there). We decompose into its local and global parts  Tglob fx   kxy , 1  xyfydy ,    And TTTloc glob. 4 6 10 Now we can apply this decomposition to our favorite Littlewood- Paley g -functions. We get the corresponding operators ggq,, loc, q glob ... for the subordinated Poisson Ornstein-

Uhlenbeck semigroup, andGGq,, loc, q glob ... for the usual Poisson semigroup.

 Proofs of 4.6.1 and 4.6.2 theorems.

128 We shall use the following known facts from[HTEV] 39

(a) gq, glob f x  Rn Q1 x,, y f y B dy 4 6 11 c where Q1 is a nonnegative kernel supported on D1 such that the associated integral operator is of weak type 1,1 and of strong type  pp,  for every p 1,  with respect to the Gaussian measure;

(b) gq,, loc f x G q loc f x Rn Q2  x, y f y  B dy , 4 6 12

Where Q2 is a nonnegative kernel supported on D2 such that: sup n Q x , y dy  And sup n Q x , y dx  . 4 6 13 x R 2 y R 2

Consequently, the integral operator associated to Q2 is of strong type for every with respect to both Lebesgue and Gaussian measures; 1 2 (c) Similar statements hold for gq and gq in place of gq .Then, using 4.5.2 theorem, we can show 4.6.1 theorem as in [HTEV]

1 . 4.6.2 theorem is dual to theorem in the case of gq , because of the general 4.3.1 theorem. Similar duality results hold

2 for and gq too. Indeed, using the facts above, we get a projection result (concerning and ) for the subordinated Ornstein-Uhlenbeck Poisson semigroup similar to 4.4.4lemma. Then we deduce the desired duality result on and .

 47 Almost sure finiteness:

49 We have seen in the previous sections (and also in [XU] ) that the Lusin cotype property is equivalent to the boundedness of the

129 various generalized Littlewood-Paley g -functions on Lp -spaces. The following result shows that this is still equivalent to an  apparently much weaker condition on the -functions.

 4.7.1 Theorem Given a Banach space B andq 2, , the following statements are equivalent: (i) Bis of Lusin co-typeq . 11 (ii) For any f LB T , Gq f z   for almost every zT . 11n n (iii) For any f LB  R,Gq f x  for almost every xR . 1 The equivalences also hold when in statement (ii) Gq is replaced by 2 1 2 Gq or Gq , and also in statement iii) Gq by Gq orGq .

49 Proof. By [XU] , 4.5.2 Theorem and 4.4.2 Corollary, we have (i)  (ii) and (i) (iii). The two converse implications are implicitly contained in [GR] 38 . Let us first prove (ii) implies (i) 1 (forGq ). To this end, observe that 1  G f z Tf zqq sup T f z   4 7 1 q      LL0,1,dr    0,1, dr          BB11rr  0   Where T is the operator that sends -valued functions to  q dr L 0,1 -valued functions given by B  1r  Pr T f z 1.  r   r  f z 4 7 2 ,1  r It is clear that T is bounded from L1 T to L1 T . B     L1 0,1 dr B   1r  Consequently, the sublinear operator f T f  is   Lq 0,1 dr B   1r 1 0 continuous from LB T  to L T  (the latter space being equipped 1 with the measure topology). By , Gq is the supremum of 1 0 1 these sublinear operators, and by ii), GLqε T for all fL B T  .

130 Therefore it follows from the Banach-Steinhauss uniform 1 0 continuity principle, Gq is continuous from LB T  to L T  . Next, we apply Stein’s theorem. A proof of the scalar version can be found in [GR] 38 , and by using the ideas there, one can prove the following vector-valued version.

4.7.1 Haar measure: 54 Definition: Haar measure is a Borel measure  in a locally compact topological group X , s.t U 0 for every non-empty Borel open set U , and  xE   E for every Borel set E .

 4.7.2 Lemma Let G be a locally compact group with Haar measure  , B be a Banach space of Rademacher type p0 and p 0 letTLGLG: B      be a continuous sublinear operator invariant under left translations. Then for every compact subset K of there exists a constant CK such that q  f Lp B x  K: Tf x   CK . 473   Withqinf p , po. In particular, if the group is compact, T is of weak type  pq, .Let us recall that every Banach space is of 1 Rademacher type1. Then, Gq is of weak type ( , ), because it is clearly sublinear and it is given by a convolution, which is invariant under translations. The proof for the implication (iii)(i) is similar. Again the 1 1 n sublinear operator ffGq  is continuous from LRB   to LR0  n . To infer as above that it is of weak type ,, we use, instead of lemma, the following.

131

 4.7.3Lemma Let B be a Banach space of Rademacher type p0 and 0pp0 .Then every translation and dilation p nn0 invariant continuous sublinear operator TLRLR: B     is of weak type  pp,  .

Remark. 4.7.1 theorem holds also for the g -function associated to the subordinated Poisson Ornstein-Uhlenbeck semigroup. In the same spirit, we also have a result similar to Theorem in the case of martingales.

4.7.4 Theorem Given a Banach space and2q , the following statements are equivalent: (i)  is of martingale cotype q . 1 (ii) If f is a martingale bounded in LB , then Sfq    almost everywhere. For the proof of this theorem, we will use martingale transform operators. Let B1and B2 be two Banach spaces, ,,F Pbe a probability space, and Fnn0 be an increasing filtration of  - subalgebras of .A multiplying sequence vv  nn1 is a sequence of random variables on  with values in L BB12,  such that each vn isFn1measurable andsup vn L .Given such n1 LBB12,  a multiplying sequence, define the martingale transform operator n T given by v by Tfn  k1 vkk d f for every martingale f . It is proved in [MT] 42 that a martingale transform operator is of weak type (,) if and only if it is of type  pp,  for1p :

132  supP Tf  C f1  Tf  Cp f p .   LL  p  0 B1 L B1 4 7 4

It is also proved there that, if T is a translation invariant martingale transform operator such that each term of its multiplying sequencevk  is a constant in L BB12, and such that f L1  Tf Converges a.e. 475 Then also verifies the inequalities in . By translation invariance of we mean that for anyk0 , the sequence kk 00 , vk, v k v k k defines a martingale transform operator Tk k1 0 0 such that for any martingale f bounded in L1 , B1 nn  Tfn  vdfk k  vdf k k k  Tf k ,1  n  B2 kk1100n B BB222 4 7 6

Now, let Qq be the martingale transform operator mapping - q valued martingales into B -valued martingales defined by the multiplying sequence v such that eachv is the constant given  k k1 k   k1  by vk  b  0, .... ,0, b ,0,.... for anybB . Then for a -valued   martingale f n q Qqn f  vkk d f d12 f, d f ,...., d ,0,... B 4 7 7 n k1 1 n q q  Q f d f, Q f  sup Q f  S f .  qnnqq k B  q  q q BBk1 n 4 7 8  Proof of 4.7.4 Theorem.

133 Given a Banach space B and2q , the following statements are equivalent: (i)  is of martingale cotype q . 1 (ii) If f is a martingale bounded in LB , then Sfq    almost everywhere.

(i)  (ii) is obvious. To prove the inverse, we use that475 implies the inequalities4 7 4, applied to Tf Qq f . Qq is translation invariant and for fL 1 1 n q q Q f Q f  d f 0 a.e. as nm,,  qqnm  1  k B B km1 479 Since it is the remaining of a convergent series (by (ii)). We end with a final remark.

49 Remark. As in [XU] for the torus, besides the Littlewood-Paley g -function we can also consider the Lusin area function on Rn . In our vector-valued setting the function is defined by 1 q q q dydt Aq  f x  x t  Pt  f y  2 n1 , 4 7 10 B t Where  x is the cone with vertex x and width 1: n1  x  t, y R : t 0, x  y  t . 4 7 11 1 Similarly, we can as well introduce the two variants Aq involving 2 only the derivative in time and Aq relative to the gradient in the space variable. As in [XU] for the torus, all the preceding results (in sections44 and46  ) are still valid withGq replaced by Aq . For instance, is of Lusin cotype q ,if and only if

134 Aq  f p n  C f LRp n for some (or all) p 1,  , if and 0nly if LR  B   1 n n fε LB  R Aq  f    a.e. on R

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