Banach Spaces and Classical Harmonic Analysis

The Erwin Schrodinger International Boltzmanngasse

ESI Institute for Mathematical Physics A Wien Austria

Banach Spaces and

Classical Harmonic Analysis

SV Kislyakov

Vienna Preprint ESI June

Supp orted by Federal Ministry of Science and Transp ort Austria

Available via httpwwwesiacat

BANACH SPACES AND CLASSICAL HARMONIC ANALYSIS

S V Kislyakov

Abstract Some p oints of contact of the two elds are discussed sp ecically pro

jections onto translation invariant subspaces Cohens theorem and related results

multipliers of H the use of invariant means psumming and pintegral op erators the

vicinity of the Grothendieck theorem some consequences of the MaureyNikishin

Rosenthal factorization theorem sets and Bourgains solution of the problem

p p

translation invariant subspaces without GordonLewis lo cal unconditional structure

Sidon sets multipliers on spaces of vectorvalued functions sp ecic spaces related to

harmonic analysis in the general theory Some pro of are indicated or even exp osed

in detail in case they are not technical and help to b etter illustrate the interplay

b etween the elds in question

Many classical Banach spaces admit a natural action of some group and many

sp ecic op erators commute with translations This features can b e used in the

study of such spaces and op erators Recipro cally sometimes the techniques of the

Banach space theory apply in harmonic analysis

This is nearly all that can b e said ab out the sub ject of this pap er in general

Furthermore the manifestations of this relationship are scarce and heterogeneous

Invoking the p olemical metaphor of Kahane and Salem for their classical b o ok

KaS on Fourier analysis il p eut ressembler en quelque sorte a un herbier

KaS Preface it can probably b e said that not only may the present text resemble

somewhat but it really is a herbarium much smaller and less systematized than

KaS representing a far less explored taxon and incomplete even in the known

part of the latter

In other words the subsequent discussion can b e viewed simply as a collection

of examples I hop e however that the reader will nd some intrinsic logic in them

and that at least sometimes he will b e amused by the interplay of the two elds

mentioned in the title

Basic denitions Throughout a group means an Ab elian group Any

group G acts on functions on it by translations f f where f y f y x

x x

x y G A linear space X of functions on G is said to b e translation invariant if

with every function f it contains all translates of f Let X and Y b e two translation

invariant spaces on G and let T X Y b e a linear op erator Then T commutes

with translations if T f T f for f X x G

x x

If G is compact which will b e assumed in what follows unless otherwise is

claimed explicitly we denote by dx the normalized Haar measure on G by the

dual group and by f the Fourier transform of a function f L G the same

notation is used for the Fourier transform of a measure Mainly we shall deal with

Supp orted in part by the Russian Foundation for Basic Research Grant no

S V KISLYAKOV

the classical spaces C G L G p M G C G Let X b e any of

these spaces and let E Then

def

ff X f for E g X

is a translation invariant subspace of X If X C G or X L G with

p all translation invariant subspaces are such Next the spaces L G and

M G are the only w closed translation invariant subspaces of L G and M G

resp ectively

In an obvious way the symb ol X makes sense for other spaces X and we shall

use this notation without further explanations

Now let each of X Y b e either L p or C and let E E

Then T commutes Y Supp ose we are given a b ounded linear op erator T X

E E

2 1

with translations if and only if it is representable in the form

T f m f

where m is a b ounded function on vanishing on the complement of E E note

that unless X Y L not every b ounded m gives rise to a b ounded op erator

via Often m is called the symbol of T A similar characterization holds if the

spaces L or M are involved under the assumption of w continuity of T

We write T T if T acts in accordance with also T is called the multiplier

with symbol m

Among sp ecic examples of translation invariant spaces we mention the Hardy

p n n n n

spaces H T where T is the ndimensional torus The dual group of T is Z

p n n n

and H T is simply the space L T p The space C T

is called the polydisk algebra the disk algebra if n All these spaces have a

wellknown interpretation as the traces on the distinguished b oundary T of certain

spaces of functions holomorphic in the p olydisk

Averaging Complementation A most usual idea in the study of translation

invariant subspaces is to average something We start with simple and old examples

further we shall come across several instances in which the realization of this idea

is more intricate

Again let each of X Y b e one of the spaces L G p or C G and

we put X let E E For every b ounded linear op erator T X

E E

2 1

T f T f dx

x x

e e

Then T commutes with translations The mapping T T is a norm pro jection

e e

and T can b e referred to as the invariant part of T Clearly T inherits many

prop erties of T but it may happ en that T

The oldest and most wellknown applications in which T is quite substantial are

related to pro jections Supp ose that X Y and E E If P is a pro jection of

e e

it is easily seen that so is P The symb ol m of P is none other than onto X X

E E

2 1

the characteristic function of E

In particular it follows that the subspace L p resp ectively C

is complemented in L in C if and only if the multiplier with the symb ol is

b ounded on L on C

BANACH SPACES AND CLASSICAL HARMONIC ANALYSIS

Prop osition For p p and G compact innite some of the

subspaces L G are uncomplemented in L G

Proof By the preceding discussion otherwise the characters of G form an uncon

ditional basis in L G From the Khinchin inequality it is easy to deduce that this

is not so if p

In the case of L G much more can b e said We refer the reader to GrMcG

for the pro of of the following statement

Theorem Cohens idemp otent theorem L G is complemented in L G if

and only if E is in the coset ring of

By denition the coset ring of is the smallest system of sets containing all

cosets of subgroups of and closed under nite unions and intersections and under

complementation

It is known that translation invariant op erators of L G or C G into itself are

precisely the op erators of convolution with a nite measure The pro jections among

such op erators corresp ond to the idemp otent measures or

So Theorem describ es also the idemp otent measures Now it is clear that the

complemented translation invariant subspaces of C G admit a characterization

similar to Theorem

A measure on G is said to b e quasiidempotent if jj jj ie for each

either or j j A set E is called a quasiGordon set if there

exists a quasiidemp otent measure such that E f g This notion was

intro duced in KwPe and will b e used later on in this pap er

Invariant pro jections in H T The subspace H T may b e complemented

in H T for some E Z not b elonging to the Bo olean ring generated by the

arithmetic progressions and singletons ie here the picture is dierent from that

describ ed in Theorem The simplest example is E f g

Theorem Paley see eg H Z For f H T we have

jf j ckf k

A similar inequality is valid for any Hadamard lacunary set E we recall that

E Z is said to b e Hadamard lacunary if E fm m g with m m

k k

for a xed constant From the prop erties of lacunary trigonometric series

is a b ounded op erator it follows that such an inequality means precisely that T

on H

Theorem see Kl The subspace H T is complemented in H T if and

only if E is in the Boolean ring of subsets of Z generated by singletons arithmetic

progressions and Hadamard lacunary sequences

Now we discuss the complemented translation invariant subspace of L G for

G lo cally compact but noncompact H P Rosenthal showed see Ro that for

such a space X the set

hX f f for all f X g

must b e in the coset ring of ie of the group dual to G and endowed with

the discrete top ology This is however rather far from a complete description We

quote a nal result for the group R see AM

S V KISLYAKOV

Theorem A translation invariant subspace X is complemented in L R if and

only if hX Z nF where F R is a nite set fx g and f g are two

in i i i i

nite col lections of reals and the numbers are pairwise rational ly independent

The ab ove general result of Rosenthal is again based on averaging and we show

how this pro cedure applies this time Let P b e a pro jection onto a translation

invariant subspace X of L G Then P X L G is a simultaneous exten

sion op erator ie P extends each b ounded linear functional on X to a b ounded

linear functional on L G and the latter is identied with a function in L G

as usual Now we dene a simultaneous extension op erator E X L G

commuting with translations as follows

hE F g i m hF P g i F X g L G

x x x

Here m is a xed invariant mean on l G ie a linear functional satisfying

m m and mf mf for f l G x G the subscript x in

the notation m shows that m is applied in the variable x For G commutative

such a functional always exists see eg Gri

Now we see that the op erator A F F E F j is a pro jection commuting

with translations and taking L G L G onto X fF L G F j

g Eventually it turns out that Cohens theorem can b e applied to deduce Rosen

thals result but some additional analysis is needed to do this The crucial step is

the observation that on the Bohr compactication G of G we can nd a measure

such that Af f for every continuous almost p erio dic function f on G Since

the dual group of G is precisely G we easily arrive at Rosenthals result with the

help of Theorem See Ro for more details

Averaging against invariant means can also b e used to show that strikingly

certain statements involving the uniform structure of a Banach space can b e lin

earized Let X b e a Banach space and Y its closed subspace Assume that there

exists a linear op erator v mapping Y to the space of uniformly continuous func

tions on X such that v f is an extension of f for every f Y Next assume that

v is continuous for the top ology of uniform convergence on b ounded sets

Theorem see Pe Remarks to x Prop osition A Under the above assump

tions there exists a bounded linear operator w Y X extending each y Y

to the entire space X

An op erator v as ab ove exists if eg Y is a uniform retract of X we may put

v f f p if p X Y is a uniform retraction We infer that if say Y is reexive

and uncomplemented in X then Y is not a uniform retract of X In particular no

innite dimensional reexive subspace of C K is a uniform retract of C K

Proof of Theorem The op erator w is given by the formula

Z Z

w f z v f x y z v f x y dy dx

X Y

R R

dx we denote some xed invariant means on l Y dy and where by

X Y

and l X X and Y are regarded as Ab elian groups under their own op erations

of addition The denition of w is consistent b ecause the integrand is b ounded

note that separately the terms in the integrand are unb ounded That w has

the required prop erties is easy The external integration is resp onsible for the

additivity of w f while the internal one for the relation w f j f Y

BANACH SPACES AND CLASSICAL HARMONIC ANALYSIS

psumming op erators Let G b e a compact space X a subspace of C G

and T X Y a psumming op erator p By the Pietsch theorem see

Basic Concepts the latter is equivalent to the existence of a probability measure

on G such that

kT f k c jf jd f X

Prop osition If G is a compact group X is translation invariant and T satises

kT f k kT f k f X x G then in we can take the Haar measure of G as

This is again done by averaging we substitute f for f in and integrate in x

against the Haar measure

The condition on T in Prop osition is fullled if say Y L G or Y C G

and T commutes with translations

Prop osition Let p and let E The identity embedding i

C G L G is pintegral if and only if the multiplier with symbol is bounded

E E

on L G

Proof The if part is clear We prove the only if part in the case where p

only slight complications arise for p but we do not dwell on this If i is

pintegral we have the following factorization

G C G C G L

v u

L L

where u v are b ounded linear op erators is a probability measure and id is the

identity inclusion By the extension prop erty of L there exists an extension w of

u making the diagram commutative Thus the op erator T v id w is a psumming

extension of i to C G It is easily seen that the averaged op erator T

T f T f dx

x x

is a psumming extension of i to C G that commutes with translations Now

Prop osition shows that T acts in fact from L G to L G consequently the

invariant pro jection of L G onto L G is b ounded

By Prop osition we now conclude that if G is innite some of the emb eddings

G are not pintegral though clearly all of them are psumming C G L

This observation was made in Pe Trace duality then yields examples of quasi

pnuclear op erators that are not pnuclear

Grothendieck theorem This theorem says that any b ounded linear op erator

T l l is summing By the lifting prop erty of l to prove this it suces

to exhibit a single summing op erator onto l Such an example can b e provided

by harmonic analysis lo osely there is a multiplier with this prop erty This way

to the Grothendieck theorem was indicated by Pelczynski and Wo jtaszczyk around

S V KISLYAKOV

T l given by f ff g is Prop osition The operator C

n Z

summing and onto

Proof The fact that is summing is a consequence of Theorem To show that

is onto it suces to prove the estimate k xk ckxk x l which by the F

and M Riesz Theorem see eg H is equivalent to the estimate

X X

k x z pz k c jx j

n n

L T

for every analytic p olynomial p We use the classical fact that the multiplier with

maps L T to L T for every r This yields symb ol

X X

n n

x z k x z pz k k k

n L n

L T

n n

It remains to refer to the classical prop erties of lacunary series

Another curiosity of similar nature is the op erator acting by the same formula as

but dened on the space C T where E Z f g It maps C T onto

E n E

l and is r summing for every r This yields a direct pro of again via harmonic

analysis of Maureys extension of the Grothendieck theorem every op erator from

l to l is r summing for every r See the survey Ki for the details

Invariant Grothendieck theorem We may state the Grothendieck theorem

like this

If H is a Hilb ert space then for every op erator u H C K the adjoint u is

summing

The problem of replacing C K in this statement by some other spaces has

received considerable attention In particular it is reasonable to ask ab out the

spaces of the form C G that verify this theorem We mention some cases in

which this is indeed so

i n E is a set for some p equivalently the space L is reexive

sets will b e discussed later in this article

ii G T E Z then C G is the disk algebra

n n n

iii G T E Z n Z

iv G T E Z f g

k k

v G T E Z

Statement i is due to the author and Pisier see eg DJT Pi for the pro of

More generally if Y is any reexive subspace of L K then the space Y

ff C K g q u for all g Y g veries the Grothendieck theorem so that

translation invariance is in fact irrelevant here Statement ii is a celebrated result

of Bourgain the pro of of which is based on ne Hardy space theory see GaKi Ki

for more information and references Statement iv is easy mo dulo ii see the

survey Ki Statements iii and v were proved resp ectively in X and Ki

Curiously the arguments are somewhat similar They are based on the ideas of the

pro of of ii but involve some additional techniques real variable Hardy spaces

Littlewo o dPaley decomp ositions etc

We see that in spite of the accidental involvement of translation invariant

subspaces the metho ds leading to iv can hardly b e classied as deserving a

BANACH SPACES AND CLASSICAL HARMONIC ANALYSIS

more detailed exp osition in this pap er at least they deviate to o much towards

only one of the two items mentioned in the title whereas the pap er is ab out the

vicinity of the conjunction and

The situation b ecomes dierent if we restrict ourselves to op erators commuting

with translations

Denition see KwPe A set E is called a Marcinkiewicz set resp ectively

a quasiMarcinkiewicz set if the multiplier with symb ol is of weak typ e

resp ectively there exists a b ounded function x on such a that for

E j j for E and the multiplier with symb ol is of weak typ e

We remind the reader that an op erator T is of weak typ e if

measfjT g j tg ct kg k t

for every g in the domain of T

Theorem see KwPe If E is a quasiMarcinkiewicz set then for every oper

ator u L G C G commuting with translations the adjoint u is summing

The pro of will b e given in Section

It is well known that Z is a Marcinkiewicz set for G T but then Theorem

gives much less than statement ii ab ove The theory of martingale transformations

see eg Z yields many examples of Marcinkiewicz sets in the dual of the group

Gu Let for instance k for all n the dyadic group D We agree

that Z f g written multiplicatively Let b e the j th co ordinate function

of D then all characters of D except the function are of the form

j j j

N 2 1

where N j j j Fixing an arbitrary set B of integers

consider the set E of the characters for which j B in the ab ove representation

Then E is a quasiMarcinkiewicz set

It is interesting that Theorem can b e adapted to the space of smo oth functions

k l

C T Neglecting the constant we may endow this space with the norm

kf k max jD f j

jsjk

equivalent to the standard one as usual D denotes the parial derivative corre

k l

sp onding to a multiindex s Thus in a natural way C T can b e regarded as

a subspace X of

l l

C T C T

the numb er of summands is equal to card fs jsj k g and X is invariant under

simultaneous translation on all copies of T Moreover the orthogonal pro jection

l l

onto the closure of X in L T L T is of weak typ e

In KwPe it was shown that using this pro jection it is p ossible to work with

X as if we had a space of the form C G with a quasiMarcinkiewicz set E

The reader may trace this insp ecting the pro of in Section In particular it

l k l

follows that any op erator u L T C T commuting with translations has

summing adjoint See KwPe for the details

S V KISLYAKOV

Theorem is proved by twofold averaging The rst averaging yields a useful

lemma stated b elow Let each of X and Y b e either L G or C G mayb e with

dierent E s let T X Y b e an op erator commuting with translations and let

S Y X b e a nite rank op erator

e e

S f dx Lemma We have trace TS trace T S where S f

x x

Indeed let b e the translation f f By the invariance of T we have

x x

trace TS trace T S trace T S

x x

and it suces to integrate in x over G

Now we remind the reader that the trace duality identies the ideal of op erators

with summing adjoint as the dual of the ideal of L factorable op erators see

DJT In particular for u L G C G we have

u supftrace uv v C G L G is of

nite rank and v g

Here as usual stands for the norm on By Lemma if u commutes with

translations the supremum can b e restricted to v commuting with translations

We see that it suces to estimate the nuclear norm v in terms of v for any

such v v c v We show a more general statement without the nite rank

assumption

Theorem If E is a quasiMarcinkiewicz set then every factorable operator

v C G L G that commutes with translations is nuclear

Proof Again we use an averaging pro cedure but this time it is intricate Let T

b e a multiplier of weak typ e the symb ol of which satises for

E and j j for E Clearly T maps L G to L G Next since

v is L factorable from the Maurey extension of the Grothendieck theorem see

the end of Section we deduce that v is summing Writing the denition of a

summing op erator in the integral form we see that if is a measure space

and F C G is a reasonably go o d function then

Z Z

kv F k d v supf jhF ij d M G kk g

As we cho ose the group G with the Haar measure and we put F

T f G where f is a trigonometric p olynomial a nite linear combination

of characters on G Clearly F indeed maps G to C G and we obtain using

invariance

jhT f ij d M G kk g kv T f k v supf

Putting e e and using the fact that T maps L to L we see that

Z Z

jhT f ij d jT f j C kf k C kf k

G G

BANACH SPACES AND CLASSICAL HARMONIC ANALYSIS

It follows that v T is a b ounded op erator from L G to L G Now let b b e the

symb ol of v We have

jf j jb f j C

for every f L G Letting f run through some approximate identity we infer

jb j that

Thus v is the convolution with an L function V the sp ectrum of which is

contained in E So v factors as follows

C G L G

wherev is again convolution with V Since id is integral andv maps to a reexive

space v is nuclear

A counterexample Returning to the b eginning of Section let us agree

to say that a space X veries the Grothendieck theorem if every op erator from l

to X has summing adjoint From the discussion in Basic Concepts see DJT

for more details it can b e seen that this happ ens if and only if every op erator

from X to l is summing From ii and iv in Section we see that the spaces

T and C T where E Z f g verify the Grothendieck theorem C

E n Z

However their injective tensor pro duct do es not

Indeed this injective tensor pruduct is none other than C T and the

latter space contains a complemented copy of l then surely the pro jection to this

g To copy is not summing This copy is spanned by the functions fz z

exhibit a pro jection we observe that the characteristic function of fk k k

Zg Z is the Fourier transform of a measure on T T The required pro jection

is given by convolution with this measure

and C are of cotyp e see eg the It is known that the dual spaces of C

E Z

survey Ki The ab ove construction can b e adapted to show that the pro jective

tensor pro duct of two cotyp e spaces may fail to b e of cotyp e moreover it may

contain a complemented copy of c

The conjectures disproved by the ab ove were p erceived as natural for some time

in the past A more radical answer to these and many other questions is given

by Pisiers celebrated counterexamples see eg Pi I have included the ab ove

material rst published in Ki b ecause of its transparence and relevance to

harmonic analysis

Stein theorem Seemingly in the pro of of Theorem we did not use the

fullscale assumption that E is a quasiMarcinkiewicz set we needed only the

L L continuity of the corresp onding multiplier The result of Stein stated

b elow shows that this is not so We denote by S G the space of all measurable

functions on G endowed with the metrizable top ology of convergence in measure

S V KISLYAKOV

Theorem Let and let T L G S G be a continuous linear

operator commuting with translations Then T is of weak type

Proof The p oint is that something similar can b e said if T do es not necessarily

commute with translations Sp ecically by the Nikishin theorem see Ma in the

latter case for every we can nd a set G with jG n j such that

jft jT f tj gj C kf k f L

Now if T do es commute with translations it only remains to average We x

eg and put Substituting f for f x G and using invariance

we rewrite in the form

C kf k

fjT f jg

integrating in x over G we get

jj jfjT f j gj C kf k

G norm sets Let p A subset E of is called a set if in L

p p

convergence is equivalent to convergence in measure

p q

This is the same as if we demand the equivalence of the L and L norms on

L for some q p

A subspace X L where is a nite measure if reexive if and only if the

norm convergence on X is equivalent to the convergence in measure see KadPe

G is reexive Thus E is a set if and only if L

The most wellknown examples of sets for all p at once are Hadamard

lacunary sets of integers here of course we have in mind the circle group T as G

In general for a xed E the set fp E is of typ e g is an interval In the pioneering

pap er Ru of Rudin it was shown that for any integer n there is a set not of

typ e for any The main idea of the construction is that j a j

expands explicitly as a linear combination of pro ducts of the characters in E and

their complex conjugates so that we may play with arithmetic conditions on E

to ensure huge cancellation after integration This leads to nontrivial examples of

sets some of which turn out to b e not of typ e

n n

The question as to whether similar examples exist for p n n n Z was

often referred to as the problem

For p the solution of this problem has turned out to b e easy at least

formally Namely if p then every set is a set for some As in

p p

the preceding section the reason is in the RosenthalNikishinMaurey factorization

theorem saying in particular that if X is any subspace of L with p on which

the top ologies of L and S are equivalent then after a change of density X b ecomes

a subspace of L with some r p

Z Z

r p r p

x x

C x X a a

p p

a a

where a is a p ositive weight a See Ma for the details Now if X is

translation invariant we easily get rid of the density by averaging which implies

BANACH SPACES AND CLASSICAL HARMONIC ANALYSIS

our claim This observation was made in BE However in reality sotosay no

sp ecic examples of sets with p that are not sets seem to b e known

this is the precise meaning of the ab ove remark on the formality of the solution in

question

For any p Bourgain Bo proved the existence of sets that are not of typ e

Thus the problem remains unresolved for p only Bourgains pro of

p p

is probabilistic and combinatorial in nature Technically it has little in common

either with harmonic analysis or with Banach space theory though philosophically

the result may b e linked with the latter

It is a known fact of the nitedimensional theory of Banach spaces that if X

is a space of dimension n and its cotyp e p constant is at most C then a typical

subspace of X of dimension C n is distant from the Hilb ert space of the same

dimension Here C dep ends only on C See MiSch Theorem Thus in any

ndimensional subspace X of L p there are many nearly Hilb ertian subspaces

of dimension prop ortional to n Now supp ose that X is spanned by characters

If we succeed in nding a subspace Y of X also spanned by characters

p p

of dimension roughly n and such that the L norm is equivalent to the L norm

on Y with a constant indep endent of X we are very close to the desired example

In other words we need to know that subspaces with typical b ehavior o ccur

even among quite sp ecic spaces namely among those spanned by characters

To understand why this nitedimensional statement suces we note that in

many cases the exp onent p in the ab ove discussion is optimal If we restrict

ourselves only to the spaces X having this prop erty then on the ab ove Y the b est

p p

constant of equivalence of the L and L norms must tend to innity as n

for every Then we attach such spaces Y to one another to obtain an innite

set that is precisely of typ e

In Bo an innite set of this sort was constructed by considering the spaces

n n+1

span fz z g on T in the role of X the attachment pro cedure consisted

in applying the Littlewo o dPaley decomp osition This is easy It is the ab ove

nitedimensional statement that is really dicult Bourgain showed that in fact

translation invariance is irrelevant in it

Theorem see Bo Let be mutual ly orthogonal functions on a

probability space let j j for al l i and let p Then there is a

subset S f ng of cardinality at least n satisfying

X X

k a k C p ja j

i i p i

iS iS

for al l scalar sequences fa g The constant C p depends only on p

In fact in a due probability sense most subspaces S of f ng p ossess the

required prop erty

Later Talagrand gave a dierent pro of of this remarkable theorem see T

Talagrands pro of is somewhat simpler than Bourgains but it leans up on ner

probabilistic background Also there is some more philosophy of Banach space

geometric nature around Talagrands pro of We quote an intermediate statement

o ccurring in T In the case of X L it is weaker than Theorem however it is applicable in a more general setting

S V KISLYAKOV

The norm of a Banach space X is said to b e smo oth if kx y k kx y k

C ky k with C indep endent of x and y It is known that for p the norm of

L is smo oth

Now let fx g b e a collection of vectors in a real Banach space X and let

i in

supf f x f X kf k g

Talagrand showed that if the norm of X is smo oth and m n

is an arbitrary xed numb er then there exists a subset I of cardinality m in

f ng such that

X X

k a x k K ja j

i i i

iI iI

In fact most subsets I of cardinality m are such Here K dep ends only on

and on the constant C in the denition of smo othness

p p

We observe that if X L then n so that the statement is close

to Theorem The reader is referred to T for the pro of of the ab ove result and

also to the pro cedure of eliminating in the case of the L norm p By the

way the metho d generalizes to some other norms an analog of Theorem for the

Lorentz space L was presented in T We do not have the p ossibility of entering

into the technicalities of either Bourgains or Talagrands pro of

Sidon sets A subset E of is called a Sidon set if jf j ckf k for

every f L G The smallest constant c is called the Sidon constant of E and is

denoted by S E Again the most wellknown example is given by the Hadamard

lacunary sequences for G T Also the co ordinate functions of the dyadic group

viewed as characters constitute a Sidon set Basically the latter sequence is none

other than the sequence of Rademacher functions on the segment

It is well known that every Sidon set E is a set for all p and

moreover

pkf k for f L G p kf k c

see eg LoR A deep result of Pisier Pi shows that in fact is a character

ization of Sidon sets

We shall discuss Sidon sets in more detail later on Here we mention only a

recent result of Kalton and Pelczynski KalPe who showed that if E is a Sidon

G is not an L space the question ab out this had b een circulating set then L

for some time b efore that In fact again the group structure has turned out to

b e irrelevant here

Theorem If is a nite measure and Q is a surjection of L onto a

space containing a copy of c then X Ker Q is not complemented in X and

the Grothendieck theorem fails for X in the sense that there is an operator from

X to l that is not summing Consequently X is not an L space and is not

isomorphic to a Banach lattice

See KalPe for the pro of and related results

BANACH SPACES AND CLASSICAL HARMONIC ANALYSIS

GordonLewis lo cal unconditional structure We refer the reader to

Basic Concepts for the precise denition recalling only that a Banach space X

p ossesses GordonLewis lo cal unconditional structure GL lust if and only if

X is a complemented subspace of a Banach lattice Harmonic analysis provides

many examples of spaces without lo cal unconditional structure

Theorem Kwapien and Pelczynski KwPe If E is a quasiMarcinkiewicz

set and there is a bounded function l E such that the multiplier T maps L

then C G fails to have GL lust to L

Theorem Pisier If E is a set then E is a Sidon set if and only if C G

has GL lust

Theorem Pisier Let E be a set and let p Then E is a set if

and only if L G has GL lust

Remark The only if parts of Theorems and are trivial b ecause C l E

for every Sidon set and L l E for every set if p

The pro ofs will b e given in the next section along the lines of the pap er KwPe

Here we make some comments In Theorem let G T E Z Then E

is a Marcinkiewicz set As we may take either see Theorem or

f ng

the function n n n Z then the continuity of T H H

is a consequence of the classical Hardy inequality n jf nj ckf k

f H see H So we recover the wellknown fact that the disk algebra fails to

have GL lust

Also compare with Section the pro of of Theorem can b e adapted to show

l k

that the space C T do es not have GL lust if k and l Clearly it

suces to prove this only for l k Then the role of T in Theorem can

b e played by the Sob olev emb edding op erator W T L T In Section it

was already explained that C T is similar to a space of the form C G with

E a quasiMarcinkiewicz set The adjustment of the details is left to the reader or

see KwPe

Passing to applications of Theorem we take E f n g Zthis is a

Sidon set for T and consider its square

k l

E f k l g Z

From condition for E it is easily seen that E is a set for T for every

p However it is also easy to observe that in the analog of for E the

constant grows as p ie faster than p Thus E is not a Sidon set By Theorem

T is isomorphic to the T fails to have GL lust we note that C C

E E

1 1

pro jective tensor pro duct of l by itself

The domain of applicability of Theorem is outlined in Section

We refer the reader to KwPe for more examples

Theorems and are proved by similar metho ds For Theorem we

consider the following op erators

C G L G C G L G

E E

Here is an arbitrary function in l E By Theorem the adjoint T is

summing If C G has GL lust the identity emb edding id factors through L E

S V KISLYAKOV

b ecause id is summing see DJT So we have come across a comp osition of

op erators b elonging to mutually adjoint op erator ideals hence it follows that

T id T ckT k k k

l E

stands for the nuclear norm or

j jj j C k k

l E

This contradicts the condition l E

For Theorem we consider the following op erators

id k T

C G L G L G C G

E E

Here k is the formal identity k is continuous b ecause E is a set Next T

is the convolution with a function x C G We claim that T is summing

Indeed T is an op erator of convolution with xt convolving this function with

a measure we obtain a function in C G which do es not dep end on a particular

choice of this measure in a xed coset mo dulo C G Hence T acts as follows

k id

1 1

L G L G T C G C G

E E

where as in id is the identity emb edding and k is a formal identity thus T

is summing

Now in id is summing so that if C G has GL lust as in the preceding

pro of we obtain

k id T C kxk x C G

jx j C kxk x C G

Thus E is a Sidon set

For Theorem we let q p and f L G Then regarded as a

mapping from L G to L G the op erator S of convolution with f factors as

follows

id k

G L G L G L G S L

E E E

Here is again the op erator of convolution with f and k is the formal identity

We see that S is summing If L G has GL lust then S factors through

L for some measure Next for p every op erator from a subspace of L

to L factors through a Hilb ert space H see Pi Chapter This implies the

following factorization for S

u v w

G H L L G S L

By the Grothendieck theorem w is summing Hence w v is Hilb ertSchmidt with

Hilb ertSchmidt norm not exceeding C kf k Since w v is also Hilb ertSchmidt

with the same norm it follows that

kS k C kf k f L G

or equivalently

jf j C kf k f L G

A simple duality argument shows that the latter condition is equivalent to the fact

that E is a set p

BANACH SPACES AND CLASSICAL HARMONIC ANALYSIS

QuasiCohen sets v s quasiMarcinkiewicz sets It turns out that in

Theorem the condition of the existence of admits a nice reformulation See

the end of Section for the denition of a quasiCohen set

Theorem Kwapien and Pelczynski KwPe The fol lowing properties of a set

E are equivalent

i E is a quasiCohen set

G we have l E G L ii For every multiplier T L

E E

iii There is a constant K such that for every trigonometric polynomial p satis

p K kpk fying p for E we have

Proof iii Let b e a measure satisfying for E j j for

E and let S b e the op erator of convolution with Then T S maps L G to

L G ie

j f j c jf j f L G

Letting f run through some approximate identity we infer that

X X

j j j j j j c

E E

iiiii Let p for E We intro duce the multiplier T

pjE

we write G To estimate the norm of this op erator for f L G L L

E E E

p jf j f x p f xdx kf k kf pk kf k kpk

hence kT p C kpk k kpk From ii we deduce that

pjE

iiii In the space C G we intro duce two convex set

W fp C G p for E p K g

U fq q kq k q for E g

Here K is the constant o ccurring in statement iii

Clearly W is closed and U is op en Also W U Indeed if p W and

p q q as in the denition of U then p q j and

X X

p q K kp q k K kq k K p K

E E

Thus there is a signed measure on G such that

Re pxdx K for p U

Re pxdx K for p W G

S V KISLYAKOV

We show that the measure dened by A A has the desired prop erties

First if E then K W hence we see that

K Re K d K Re

ie j j Second if E and is any complex numb er then p K U

b ecause p for E Thus K Re K and

Now we can restate Theorem as follows if E is a quasiMarcinkiewicz set but

not a quasiCohen set then C G fails to have GL lust

Sidon sets and arithmetic diameter For a nite set its arithmetic

diameter d is dened as the smallest N such that C G is at most distant

relative to the BanachMazur distance from a subspace of the N dimensional

space l If E is a Sidon set in then C G is at most S E distant from l

for every nite E Consequently the arithmetic diameter of must grow

exp onentially as a function of jj

There are many ways to see this For instance we may argue as follows First

recall that for p the typ e constant of L is of order c p Next it is easily seen

log N

that the BanachMazur distance b etween l and l is b ounded uniformly in N

log N Finally if e e are so the typ e constant of l do es not exceed c

k r te kdt the co ordinate unit vectors in l then the Rademacher average

i i

n Thus if l is is equal to n so that the typ e constant of l is at least

n n

emb eddable in l then n C log N as desired

It is remarkable that the exp onential dep endence of d on jj in fact charac

terizes the Sidon sets

Theorem Bourgain Bo If log d jj for every nite E then E

is a Sidon set Moreover we have S E c where c is a universal constant

The pro of of this Banachgeometric characterization of Sidon sets can hardly b e

called geometric rather it is combinatorial b ecause the only geometric notion

involved is that of entropy In general in a metric space with metric the

entropy of a set F is the logarithm of the smallest cardinality N N F of

an net for F

The idea of using entropy in the theory of Sidon sets was originally exploited

by Pisier see eg Pi Later Bourgain revised Pisiers work on Sidon sets

replacing some ne probability metho ds by an elementary random choice combined

with entropy combinatorics and with harmonic analysis arguments We refer the

reader to the Bourgains survey Bo and to the references therein for this The

following statement is Bo Corollary this is a slight improvement of Pisiers

original entropy characterization of Sidon sets see Pi p For a subset of

we dene a metric on G as follows

x y sup j x y j

Theorem Let E If for every nite set E and some we have

G N

BANACH SPACES AND CLASSICAL HARMONIC ANALYSIS

then E is a Sidon set and S E C log

We show how Theorem is deduced from Theorem Supp ose that E is

such that log d jj for all nite sets E For any nite along with

we intro duce another greater metric on G

x y supfjf x f y j f C G kf k g

It is easily seen that for the arithmetic diameter d we have d N G

Indeed denoting the latter quantity by N we nd a net x x for G in

the metric Then the mapping f f x f x is a go o d emb edding of

C G into l

Next xing a nite set E we clearly have

B B

x y S x y

for all B yielding

jB j

N B G dB N B G

Now Theorem implies the inequality

S C logS

which do es not allow the quantity S to grow innitely as expands More pre

cicely we obtain S C for every nite E whence S E C

Sidon sets and cotyp e

Theorem Bourgain and Milman see BoMi Bo If C G is of cotype q

for some q then is a Sidon set

Conversely if is a Sidon set then C G l which is of cotyp e

Recalling the wellknown characterization of the spaces with nite cotyp e we

can restate Theorem in the following way either is a Sidon set or C contains

the spaces l uniformly

Theorem can b e deduced from Theorem and the following subtle fact of

the theory of nitedimensional Banach spaces

Lemma Let X be an ndimensional normed space isomorphic to a subspace

of l Then n C C X log C X log N

Here C X r is the cotyp e r constant of X

We refer to Bo for the pro of of this lemma

Now we prove Theorem Let E and let C b e of cotyp e q ie

C C G We x a nite set E and obtain an a priori estimate for S

q E

like that at the end of the preceding section If B then C C G S

By Lemma

q q

jB j C C C G log S log dB

q E

where as b efore dB stands for the arithmetic diameter of B Now Theorem

implies that

S C C C G log S

q E

S V KISLYAKOV

which do es not allow the numb ers S to b e unb ounded Thus E is a Sidon

set

It should b e mentioned that the case of q in Theorem is less involved this

case had b een analyzed indep endently by Pisier and by Kwapien and Pelczynski

prior to Theorem We can argue nearly as in the pro of of Theorem Indeed

let C G b e of cotyp e For a function x C G the op erator of convolution

E E

with xt can b e viewed in a natural way as a mapping from C G to C G

E E

So the domain of is the conjugate of a cotyp e space and its range is a cotyp e

space By Pisiers factorization theorem see Pi Chapter factors through

a Hilb ert space H This factorization is shown in the following diagram

L G C G C G L G

E E

Here j and k are identity emb eddings Now k is summing hence k is Hilb ert

Schmidt Considering the adjoints in a similar way we see that j is Hilb ert

Schmidt Concequently the comp osition k j is nuclear with k j C kxk

Since this comp osition is again the op erator of convolution with xt we obtain

jx j C kxk

Multipliers on spaces of vectorvalued functions Let X b e a Banach

space then the spaces of X valued functions L G X p or C G X

are dened as usual As in the scalar case for E the spaces L G X and

C G X are distinguished by the condition

def

f for E f

Now assume we are given a b ounded multiplier of scalar spaces T L G

L G acting in accordance with formula The expression m f makes sense

also for X valued functions f so it is natural to ask ab out the description of the

p p

1 2

spaces X for which generates a b ounded op erator from L G X to L G X

E E

1 2

In fact for each particular multiplier T this question presents a mystery which

can b e claried only rarely and by dissimilar techniques Below we briey discuss

two imp ortant cases Beyond this we mention the pap er BlPe in which the vector

valued analogs of the Paley inequality see Theorem and the Hardy inequality

n jf nj C kf k f H T

as well as some related questions were discussed In spite of a quite considerable bulk

of information presented in BlPe no complete description of the corresp onding

classes of Banach spaces X is available in these cases as yet

B convexity and K convexity Here our basic multiplier is the orthogonal

pro jection of L D D is the dyadic group see Section for the denition onto

the subspace generated by the co ordinate functions of D Identifying D with

in the usual way we arrive at the Rademacher pro jection The spaces X for which

this pro jection extends in a natural way to L D X are said to b e K convex

The notion of K convexity is quite useful in the theory of Banach spaces pri

marily due to the fact that it is intimately related to the duality b etween typ e and

cotyp e Remarkably K convex spaces admit a complete characterization

BANACH SPACES AND CLASSICAL HARMONIC ANALYSIS

Theorem Pisier see Pi A Banach space X is K convex if and only if X

does not contain the spaces l uniformly

The spaces with the latter prop erty are called B convex A space is B convex if

and only if it is of nontrivial typ e see Basic Concepts

UMDspaces Here our basic multiplier is the Hilb ert transformation H on

L T We have H T where mk i sgn k k Z

It is really quite useful to know for which Banach spaces X the Hilb ert transfor

mation more generally an arbitrary CalderonZygmund singular integral op erator

acts on the L space of X valued functions The general theory reduces the case of

any p to the case of p see eg St

Surprisingly the class of spaces X for which H acts on L T X admits a com

plete description

Theorem Burkholder McConnel Bourgain A Banach space X has the above

property if and only if there is a biconvex function X X R such that

and x y kx y k if kxk ky k

The same condition is equivalent to the continuity of standard martingale trans

formations on L X That is why the spaces X such that H acts on L X are

called UMDspaces UMD is for unconditionality of martingale dierences The

pro of of Theorem is indirect and passes via this statement on martingales We

refer the reader to Burkholders survey Bu and to the references in it for the

details

It should b e noted that the condition formulated in Theorem is dicult to

work with Basically the only way to prove the existence of the ab ove on a Banach

space X is to verify the continuity of H or of the martingale transformations on

L X directly The Hilb ert space seems to b e the only one presenting a simple

p ossibility of exhibiting for instance we may put x y Rehx y i

Direct verication of the continuity of H shows that the spaces L with p

and the Shattenvon Neuman classes C again with p are UMD

spaces Next the prop erty of b eing a UMDspace is inherited by the subspaces

and quotient spaces This is a sup erprop erty if X is nitely representable in Y

and Y is UMD then so is X A UMDspace must b e sup erreexive but not

all sup erreexive spaces are UMD As b efore we refer the reader to Bu and the

references therein Again the pro ofs of the ab ove statements do not use Theorem

The ab ove discussion should b e supplemented with the following result due to

Bourgain

Theorem If is a nite measure and r then the Hilbert transforma

tion H is bounded from L T L to L T L

We refer the reader to the survey Ki for the pro of and related material The

result can b e used to verify statement ii in Section This idea is Bourgains the

details of this verication can also b e found in Ki

Returning once again to statements iv in Section we note that we may

ask any Banach space theory question ab out any sp ecic space arising in harmonic

analysis This will yield an incontestable p oint of contact of the two elds but

rarely will this show a real interplay b etween them In many cases a pure problem

S V KISLYAKOV

of hard analysis even without the adjective harmonic arises in this way as is

describ ed in Section after statements iv

Some exceptions of these rule were however discussed ab ove Another one

is presented by the still mysterious space U of uniformly convergent Fourior series

on the unit circle Bourgain was the rst to prove that U is weakly sequentially

conplete and his pro of involved dicult techniques of hard analysis see Bo But

later it was discovered that the statement can b e veried almost entirely within the

soft metho ds of functional analysis We refer the reader to the pap er GaKi in

this collection for a discussion and references

Having mentioned the space U we probably cannot avoid considering the spaces

of trigonometric p olynomials on the circle Let

p n

P spanf z z g

with the metric of L T p In a way and from the harmonic analy

sis viewp oint these spaces may b e regarded as elementary n dimensional

blo cks building the Hardy classes H T For p this is emphasized by

p p

the fact that the P are complemented in H uniformly in n However for p

p p

or p the norm of the invariant pro jection of H onto P grows as c log n and

by averaging the norm of any other pro jection cannot b e smaller The following

observation folklore was made by Bourgain and Pelczynski ab out years ago

Prop osition The spaces P respectively P can be embedded uniformly

n n

complemented ly in H respectively H

Proof We treat only the case of H the other one b eing similar Considering the

subspaces generated by the o dd or by the even p owers of z it is easy to deduce

H H that H H H Thus instead of H we emb ed P into z

T The emb edding in question is given by the formula T L L

Z Z

I p z p p p P

Now we dene an op erator J L L P by the formula

Z Z

J f g z K f K g

n n n

where K is the nth Fejer kernel It is easily seen that J I id

n n n

Without entering into the details we refer the reader to Bo for a construction

of a go o d basis in P and to GoR for evaluation of various constants such as

GL lust BanachMazur distances to various spaces etc for the spaces P

Let us stop at this p oint

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BANACH SPACES AND CLASSICAL HARMONIC ANALYSIS

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