Real and Complex Operator Norms

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2007:9 LICENTIATE T H E SI S

Natalia Sabourova

Luleå University of Technology Department of Mathematics

2007:9|: -1757|: -lic -- 07⁄9 --  Real and Complex Operator Norms

Natalia Sabourova

Department of Mathematics Lule˚a University of Technology SE-97187 Lule˚a Sweden

March 2007

Abstract

Any bounded linear operator between real (quasi-)Banach spaces T : X → Y has a natural bounded complex linear extension TC : XC → YC defined by the formula TC(x + iy)=Tx + iT y for x, y ∈ X,whereXC and YC are so called reasonable complexifications of X and Y , respectively. We are interested in the exact relation between the norms of the operators TC and T . This relation can be expressed in terms of the constant γX,Y appearing in the inequality ≤ TC γX,Y T considered for all bounded linear operators T : X → Y between (quasi-)Banach ≤∞ spaces. The work on the constant γLp,Lq for 0 norm Lebesgue spaces. Looking for the criteria of the equality of real and complex norms of operators from a Banach lattice into the same Banach lattice we find a number of examples of two dimensional Orlicz spaces different from Lebesgue spaces and a simple operator between them with non-equal real and complex norms. We also consider in detail the Clarkson inequality which can be interpreted in terms of a certain operator norm inequality appearing as an example in many parts of the thesis. It turns out that complex norm of this operator can be easily obtained but to find the real one is not so trivial. With the help of the Clarkson inequality we construct an operator between Lebesgue spaces with non- atomic measures which has different real and complex norms. Finally, we consider both complex and real versions of the Riesz-Thorin interpolation theorem in the first quadrant and by using numbers γp,q find, for example, that the real Riesz constant is bounded by 2 for all 0

Acknowledgement

I am strongly indebted to Professor Lech Maligranda for his assistance and numerous valuable remarks and suggestions without which this thesis could not be written. I express my gratitude to my co-supervisor Boris Postnikov for his warm encouragement. I am also grateful to Natalia Goza, Gunnar Söderbacka and Per Leander for their friendly support. I thank the Swedish Institute for financial support of my studies in Sweden and Wallenbergsstiftelsen “Jubileumsanslaget” for travel grant that made it possible for me to visit a conference in Poland. I am also very thankful to my parents and my husband Thorbjörn Wikström for their constant care and love.

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CONTENTS

Chapter 1. Introduction 1 Chapter 2. Complexification of Real Spaces and Operators 7 2.1. Complexification of real linear spaces and operators 7 2.2. Complexification of real normed spaces 8 2.3. Complexification of real Banach lattices 11 2.4. Complex norms of operators 12 Chapter 3. Equality of Real and Complex Operator Norms 13 3.1. Operators between Lebesgue spaces 13 3.2. Operators between spaces of bounded p-variation 19 3.3. Operators between Banach lattices 24 3.4. Operators between Orlicz spaces 26

Chapter 4. Basic Properties of Constant γX,Y 29 Chapter 5. Clarkson Inequality in the Real and the Complex Case 37 5.1. The best constant in the complex case 37 5.2. The best constant in the real case 39 5.3. Some applications of the Clarkson inequality 44

p q Chapter 6. Basic Properties of Constant γ(ln,lm)51 Chapter 7. Riesz-Thorin Interpolation Theorem 61

CHAPTER ONE

Introduction

Any bounded linear operator between real (quasi-)Banach spaces T : X → Y has a natural complex extension TC : XC → YC deﬁned by the formula

TC(x + iy)=Tx+ iT y for x, y ∈ X.

Now we just say that the complex (quasi-)Banach spaces XC and YC are obtained from X and Y by some reasonable complexification procedure that we will explain in the next chapter. We call the norms of the operators T and TC as real norm and complex norm, respectively. In this thesis we study the relation between complex and real norms of arbitrary bounded linear operator T between two real (quasi-)Banach spaces or, more precisely, we study the constant γX,Y in the inequality ≤ TC γX,Y T . The main reason for us to consider this relation is that it allows to interpret the results obtained for complex operator norms in the real settings as it is, for example, in the Riesz-Thorin interpolation theorem. Despite it is enough to have some good estimate of the constant γX,Y for doing such interpretation we are also interested in the techniques of finding the best constant γX,Y and put special emphasis on the case when γX,Y =1, i.e. the case of equality of complex and real operator norms. Probably the first interest to the question about the relation between complex and real operator norms appeared in the paper of Marcel Riesz [50]onconvexity and bilinear forms from 1927 but published in 1926! In this paper Riesz proved his well-known convexity theorem for bilinear forms by ”real-variable” techniques p → q which can also be stated in the following way: the norm of the operator T : ln ln as a function of (1/p, 1/q) is logarithmically convex in the ”lower triangle” {(α, β) ∈ R2 :0≤ β ≤ α ≤ 1}. Besides, Riesz raised a question on the validity of this theorem for complex Lebesgue spaces in the entire unit square, i.e. in {(α, β) ∈ R2 :0≤ α, β ≤ 1}. As a starting point for the following study of this question he provided an example of an operator between three-dimensional Lebesgue spaces with different real and complex norms. Moreover, he briefly presented arguments in support of the statement about the equality of real and complex norms of operators between finite- dimensional lp spaces. The detailed proof of this statement can be found in the work of Taylor [56] (see also Crouzeix [12]). In 1939 Thorin [57], the student of Riesz, gave an affirmative answer to Riesz’s question publishing an elegant proof using an elementary convexity property of analytic functions, the so called ”three lines lemma”. In the thesis on convexity theorems [58] he pointed out that Riesz theorem

1 2 1. INTRODUCTION for real bilinear forms in the ”lower triangle” can be obtained as a consequence of the proved by him complex case. Thorin also worked out an example of Riesz showing the impossibility of extending the real convexity theorem to a convex region beyond and containing the ”lower triangle” even in the case of two-dimensional lp spaces. Thus, the importance of the relation between complex and real operator norms was recognized. Marcinkiewicz and Zygmund [44] used Gaussian variables to obtain vector-valued estimates of the operators between (quasi-)Banach Lp − Lp spaces and some years later Zygmund included a simple proof of the equality of complex and real norms for the operators between Lp[a, b] spaces in his book Trigonometric Series [65]. Verbicki˘ı and Sereda in [59] profitably employed the simple idea of Zygmund and established the equality of complex and real norms of the operators between Banach Lp(μ) − Lq(ν) spaces with arbitrary σ-finite measures μ and ν ≤ and p q. At the same time Verbicki˘ıin[60] began to consider constant γX,Y for arbitrary Banach spaces X and Y paying special attention to the possible ways of complexification of real normed spaces. He improved the general estimate of the relation between complex and real norms for the operators between Lebesgue spaces, i.e. γp,q, and found the exact value of γX,Y inthecaseofoperatorsbetween spaces of trigonometric polynomials supplied with Lp norms. Then the highly non- trivial works of Krivine [33]-[34] evolved. Using the advantage of tensor product technique Krivine found the exact value of γ∞,1, which is turned out to be the ≤ ≤∞ biggest among all γp,q in the case 1 p, q . Utilizing further the tensor product technique Figiel, Iwaniec and Pelczyński [17] established the equality of complex and real norms for the vector-valued operators from Lp into Lq space with 1 ≤ p ≤ q as well as Defant extended the range of the values p and q for which the exact constant γp,q was found. Some interest to this problem continues to appear in literature up to now. For example, Gasch, Maligranda [19] and Luna-Elizarrarás, Shapiro [36] continued the study of the relation between complex and real norms for vector-valued operators. Kirwan [31]andMuñoz, Sarantopoulos and Tonge [46] studied various procedures of complexification of real normed spaces and obtained the estimates for the complex norms of multilinear mappings and polynomials. Maligranda and Sabourova [43] closely considered real and complex versions of the classical Clarkson inequality in the content of the question about the relation between real and complex operator norms. They found an example of an operator between Lebesgue spaces with non-atomic measures with different real and complex norms. And finally we will mention the work of Holtz and Karow [23], where some of these questions appeared again. It is not surprising that the most complete answer to the question about the relation between complex and real operator norms exists only in the case of Lp spaces and for the spaces which are close to Lp although there are still some open questions even in this case. So we describe the problem in the framework of Lp spaces and then formulate it in the more general settings. In this thesis we use standard notations. For 0

p 1/p f + igp =( |f(x)+ig(x)| dμ(x)) . Ω 1. INTRODUCTION 3

If p = ∞ we have a Banach space of complex-valued essentially bounded functions with the norm:

f + ig∞ = ess sup{|f(x)+ig(x)| : x ∈ Ω}, where ess sup{h(x):x ∈ Ω} =inf{M>0:μ{x ∈ Ω:h(x) >M} =0}. When the underscript C is omitted the functions are assumed to be real-valued on Ω. Furthermore, if X and Y are arbitrary (quasi-)Banach spaces, then L(X, Y ) denotes the collection of all bounded linear operators from X into Y endowed with the supremum (quasi-)norm and L(X) replaces L(X, X). In this thesis we work only with this kind of operators. For 1

(1.1) TC(f + ig)=Tf + iT g, p where f,g ∈ L (μ) and the norm of the complexiﬁcation operator TC is given by:

TC(f + ig)q TC =sup . p,q f+igp=0 f + ig p

When no confusion can arise we simply write T instead of T p,q. For any bounded linear operator T between normed spaces with so called reasonable com- plexiﬁcations we have a general estimate of the norms T ≤TC≤2T (see Chapter 2, Proposition 3). In particular, for any operator T ∈L(Lp(μ),Lq(ν)) with 1 ≤ p, q ≤∞we obtain

TC(f + ig)q = Tf + iT gq ≤Tfq + Tgq ≤T p,qfp + T p,qgp 2 2 1/2 2 2 1/2 ≤T p,q(|f| + |g| ) p + T p,q(|f| + |g| ) p 2 2 1/2 ≤ 2T p,q(|f| + |g| ) p.

This relation establishes boundedness of the operator TC. Hence, the natural com- p q plexification TC of any operator T ∈L(L (μ),L (ν)) defined by (1.1) is a complex bounded linear operator with the norm T p,q ≤TCp,q ≤ 2T p,q and it makes sense to consider the ratio TCp,q/T p,q. Thus, we are interested in the numbers: { { ≥ ≤ ∈L p q } γp,q =sup inf γ 1: TC p,q γ T p,q for any T (L (μ),L (ν)) : for arbitrary positive σ − finite measures μ, ν}. In other words, for any T ∈L(Lp(μ),Lq(ν)) and f,g ∈ Lp(μ) 2 2 1/2 TC(f + ig)q = |Tf + iT g|q = (|Tf| + |Tg| ) q ≤ ≤ | |2 | |2 1/2 TC p,q f + ig p γp,q T p,q ( f + g ) p, means that γp,q is the best constant in the inequality | |2 | |2 1/2 ≤ | |2 | |2 1/2 (1.2) ( Tf + Tg ) q γp,q T p,q ( f + g ) p for all f,g ∈ Lp(μ)andallT ∈L(Lp(μ),Lq(ν)). ≤ ≤ The above estimates show only that 1 γp,q 2. We note however, that linearizing the expression under the norm sign on the left hand side of inequality 4 1. INTRODUCTION

(1.2) in terms of a certain integral and using the possibility to exchange the order of taking the operator norm and integration, we could extract the norm of the real operator T and following the same procedure we could get back the expression on the right hand side. As a result of this procedure we could ﬁnd a better estimate of ≤∞ γp,q from above. Calculation of the exact value of γp,q for all 0

(n) { p q − } Kp,q (r)=sup KL (μ),L (ν)(r): foranypositiveσ ﬁnite measures μ, ν , where n n (r) { ≥ | |r 1/r ≤ | |r 1/r KLp(μ),Lq (ν)(n)=inf C 1: ( Tfk ) q C T ( fk ) p, k=1 k=1 p p q for all f1,f2, ..., fn ∈ L (μ)andT ∈L(L (μ),L (ν))}.

(r) In other words, constants Kp,q (n) establish the biggest possible relation between p r → q r the norm of a natural vector-valued operator extension TE : L (μ, ln) L (ν,ln) of the operator T defined by TE(f1, ..., fn)=(Tf1, ..., T fn) and the norm of the (2) operator T . Thus, with the preceding notations γp,q = Kp,q (2) and in this thesis we will focus on these constants connected to Lp spaces. The discussion around the complexification of Lebesgue spaces and operators between them can be completely adapted to arbitrary Banach lattices X and Y and as a consequence the same kind of inequality as (1.2) will define the constant γX,Y (for explanation see Chapter 2). For arbitrary (quasi-)Banach spaces X and Y the constant γX,Y is no longer necessary connected to inequality (1.2) as it is, for example, in the case of operators between spaces of functions of bounded p- variation. However, even in this case we will systematically use, when it is possible, the same tool, namely the linearization of the expression (a2 + b2)1/2 in terms of a certain integral and the possibility to exchange the order of taking the operator norm and integration. This thesis is organized as follows. In Chapter 2 we consider the question about ”natural” complexifications of real (quasi-)normed spaces and define the notion of reasonable norm which will play a crucial role in getting that the relation between complex and real norms of operators between arbitrary Banach spaces is bounded by 2. Chapter 3 is about the equality of real and complex operator norms. We consider the operators between Lebesgue spaces, mixed norm Lebesgue spaces, some other Banach lattices, spaces of functions of bounded p-variation. Looking for the criteria of the equality of real and complex norms of the operators between Banach lattices we found a connection of this question with the notions of p-convexity and q-concavity of these spaces that we present here. We also show some examples of simple operators from two-dimensional Orlicz space different from Lp space into the same Orlicz space with non-equal real and complex norms. In Chapter 4 we summarize the results obtained for the constant γX,Y ,where X and Y are Lebesgue spaces, mixed norm Lebesgue spaces, spaces of functions of 1. INTRODUCTION 5 bounded p-variation. The most interesting here is that we establish the estimate of the constant γX,Y by 2 even for the quasi-Banach variants of these spaces. Chapter 5 deals with real and complex versions of the so called generalized Clarkson inequality which can be interpreted in terms of the operator norm inequality. This operator appears as an example in many parts of this thesis and sometimes we even need to know the exact values of its real and complex norms. We also present here some other applications of the Clarkson inequality, including a construction of an operator but now between Lebesgue spaces with non-atomic measures for which real and complex norms are different. We note, that this chapter was initially written as an article ”Clarkson Inequality in the Real Case” which will appear in the journal Mathematische Nachrichten. p q In Chapter 6 we summarize the basic properties of the constant γ(ln,lm)which is just a particular case of the more general constant γp,q. The main reason why this chapter appears is that we wanted to know if the constant γp,q is finite when q p q tends to zero and tried to answer the question for the more simple constant γ(ln,lm). However, the affirmative answer was obtained directly for γp,q. In spite of this fact we found it interesting to improve for some values of p, q, n, m the estimates of p q γ(ln,lm) obtained in Chapter 4 for γp,q. Finally, Chapter 7 treats how with the knowledge of the constant γp,q to get the real Riesz-Thorin interpolation theorem from its complex analogue. We consider this theorem in the whole first quadrant and what is new we show that even in the case when at least one of the spaces is quasi-Banach the so called Riesz constant is not bigger than 2.

CHAPTER TWO

Complexiﬁcation of Real Spaces and Operators

Complexification of a real normed space is taking place on two different levels. On the algebraic level we complexify the elements of the real linear space obtaining a complex linear space. Then, on the geometric level, we introduce on this complex linear space a new norm. To get a complex normed space which inherits some of the properties of the original space this new norm should be reasonable, the notion which we explain in this chapter. It turns also out that if complex normed spaces are supplied with such reasonable norms than the relation between complex and real norms of bounded linear operators between them does not exceed 2 and if the norms are not reasonable this estimate may not necessarily holds. We note, although we do not write it explicitly, that the complexification of real quasi-normed spaces follows the same procedure as we describe in this chapter for real normed spaces and the notion of reasonable quasi-norm is defined in exactly the same way as reasonable norm.

2.1. Complexiﬁcation of real linear spaces and operators

Let V be a real linear space. The natural complexiﬁcation VC of V can be constructed in the way similar to usual construction of the complex numbers from the reals. Thus, VC can be deﬁned as the direct sum

VC = V ⊕ iV = {x + iy : x, y ∈ V } for which the complex linear structure given by the relations:

(i) (x1 + iy1)+(x2 + iy2)=(x1 + x2)+i(y1 + y2),x1,x2,y1,y2 ∈ V , (ii) (a + ib)(x + iy)=(ax − by)+i(bx + ay),x,y ∈ V and a, b ∈ R.

Obviously, VC is a complex span of V. A briefly review of other possible alternative descriptions of complexification of a real linear space can be found in the paper of Muñoz-Sarantopoulos-Tonge [46]. Complexifying real linear spaces we extend linear bounded operators between them to the complex linear operators between complexification spaces. Namely, if T : X → Y is a linear operator between real linear spaces X and Y, then its complex linear extension is naturally defined by

TC(x + iy)=Tx+ iT y,

7 8 2. COMPLEXIFICATION OF REAL SPACES AND OPERATORS where x, y ∈ X. To check linearity of TC we need to establish the additivity and homogeneity of TC.Let x, y, u, v ∈ X and a, b ∈ R,then

TC(x + iy + u + iv)=T (x + u)+iT (y + v)

= Tx+ iT y + Tu+ iT v = TC(x + iy)+TC(u + iv)

TC((a + ib)(x + iy)) = T (ax − by + i(ay + bx)) = T (ax − by)+iT (ay + bx)=aT x − bT y + iaT y + ibT x

=(a + ib)(Tx+ iT y)=(a + ib)TC(x + iy).

It is worth to remark that the equality TC(x+i0) = Txallows us to identify L(X, Y ) with a real subspace of L(XC,YC).

2.2. Complexification of real normed spaces Complex extensions of real normed spaces require new norms. It is quite natural to look for the norms which replicate some basic properties of complex numbers. Definition . · · 1 Let (X, X ) be a real Banach space. A norm XC on the complexification XC is said to be reasonable if ∈ (i) x XC = x X for any x X ( complexification norm) iθ ∈ ∈ R (ii) e z XC = z XC , where z XC and θ ( symmetry of the unit ball) ∈ (iii) z XC = z XC for any z XC ( conjugation-invariant norm) Property (i) defines so called complexification norm and simply means that a · · reasonable norm XC extends the original real norm X. Property (ii) says that a reasonable norm possesses symmetry of the unit ball. Property (iii) describes a conjugation-invariant norm. These properties imply, for example, that the constructed complexified space inherits separability and completeness of the original space. Moreover, due these properties all reasonable norms on the complexified space are equivalent and, what is the most important in our work, that the relation between complex and real norms of bounded linear operators between them does not exceed 2 (cf. Holtz-Karow [23], Kirwan [31]). The proofs of all these statements are based on the following easily obtained consequence of property (iii). Namely, ∈ ≤ property (iii) implies that for all z XC it follows Re z X, Im z X z XC . ≤ Indeed, 2 Re z X = z + z XC 2 z XC and since Im z =Re(iz), then also ≤ Im z X z XC . Conversely, if property (iii) does not holds we can construct a complexification of a real space for which Re z X, Im z X > z XC as it is shown in the following example. Example 1. Let p ≥ 1 and 0 <α<(1/2)p. Consider a two-dimensional space X = C2 equipped with the norm | − |p − | |p 1/p ∈ C2 z XC =(α z1 iz2 +(1 α) z1 + iz2 ) for all z =(z1,z2) . · 2 It is easy to check that XC defines a complexification of usual norm on l2 and · thus satisfies property (i) of Definition 1. Clearly, XC also satisfies property (ii). 1/p Now, if we take z =(1,i), then Re z X = Im z X =1and z XC =2α < 1for p 0 <α<(1/2) . Hence, Re z X , Im z X > z XC . Moreover, this example also shows that property (iii) is not a simple consequence of the first two. Taking the 1/p − 1/p same z we get z XC =2α and z XC =2(1 α) . Therefore z XC = z XC since otherwise α should be equal to 1/2 which is not the case in our settings. 2.2. COMPLEXIFICATION OF REAL NORMED SPACES 9

Proposition 1(cf.Mu˜noz-Sarantopoulos-Tonge [46]). Let XC be a reasonable complexiﬁcation of a real Banach space X. For any x, y ∈ X we have − ≤ sup x cos θ y sin θ X x + iy XC 0≤θ≤2π and ≤ − x + iy XC inf ( x cos θ y sin θ X + x sin θ + y cos θ X ). 0≤θ≤2π

Proof.Property (ii) of Deﬁnition 1 implies that for any θ ∈ R we have eiθ(x + iy) = x + iy . Then, using property (iii) we conclude XC XC Re eiθ(x + iy) = Re(x cos θ − y sin θ + i(x sin θ + y cos θ)) X X = x cos θ − y sin θ ≤ eiθ(x + iy) = x + iy . X XC XC On the other hand, x + iy = eiθ(x + iy) = x cos θ − y sin θ + i(x sin θ + y cos θ) XC XC XC ≤ − x cos θ y sin θ X + x sin θ + y cos θ X . Thus, for all x, y, θ ∈ R we have (2.1) − ≤ ≤ − x cos θ y sin θ X x + iy XC x cos θ y sin θ X + x sin θ + y cos θ X . Now, taking the supremum on both sides of the left inequality and inﬁmum on both sides of the right inequality in (2.1) over 0 ≤ θ ≤ 2π we obtain the statement of the proposition.

Proposition 1 also says that among all reasonable norms the smallest one is given by the formula − x + iy T =sup x cos θ y sin θ X . 0≤θ≤2π The notation T stands for Taylor norm who first introduced this kind of reasonable norm in [45]. Another way to describe the Taylor norm is by − x + iy T =supx cos θ y sin θ X 0≤θ≤2π =supsup |ϕ(x)cosθ − ϕ(y)sinθ| 0≤θ≤2πϕ ∗ ≤1 X =supϕ(x)2 + ϕ(y)2. ϕX∗ ≤1 These two descriptions appeared after Taylor’s unsuccessful attempt to define the so called ”natural” complexification norm in [55]by 2 2 (2.2) x + iy nat = x X + y X . The problem with this norm is that, in general, it fails to have property (ii) of Definition 1 and hence it is not reasonable. It becomes reasonable only in the 10 2. COMPLEXIFICATION OF REAL SPACES AND OPERATORS

· settings of Hilbert spaces. Indeed, if the norm on (XC, nat) defined by (2.2) is reasonable, then 2 | |2 2 2 2 x + iy nat = 1+i x + iy nat = (1 + i)(x + iy) nat − 2 = (x y)+i(x + y) nat − 2 2 2 2 = x y X + x + y X =2( x X + y X ). · Hence, (X, X ) satisfies the parallelogram law and hence is a Hilbert space. Con- · versely, assume that (X, X ) is a Hilbert space, then property (ii) can be obtained from iθ 2 − 2 2 e (x + iy) nat = x cos θ y sin θ X + x sin θ + y cos θ X = x cos θ − y sin θ, x cos θ − y sin θ + x sin θ + y cos θ, x sin θ + y cos θ = x, x cos2 θ − 2 x, y cos θ sin θ + y,y sin2 θ + x, x sin2 θ +2 x, y cos θ sin θ + y,y cos2 θ 2 2 2 = x X + y X = x + iy nat · and properties (i) and (iii) hold obviously. Hence, the norm nat is reasonable in the settings of a Hilbert space. There are also some interesting observations concerning the Taylor complexification. From Proposition 1 it easily follows that all other reasonable norms · on XC are in the following relation to the Taylor norm ≤ ≤ x + iy T x + iy XC 2 x + iy T . Moreover, Taylor complexification of real Banach spaces is also a way to extend linear bounded operators between real Banach spaces to the complex ones between their complexifications without increasing the norm. Proposition 2. For any T ∈L(X, Y ),whereX, Y are real Banach spaces and ∈L · · TC ((XC, T ) , (YC, T )) we have TC = T .

Proof. Obviously, TC≥T . On the other hand,

C − T (x + iy) XC =supT (x)cosθ T (y)sinθ X 0≤θ≤2π − =supT (x cos θ y sin θ) X 0≤θ≤2π ≤ − T sup x cos θ y sin θ X 0≤θ≤2π ≤ T x + iy XC and TC≤T . Hence, TC = T . Remark 1. Thus, in the case of Hilbert spaces for which Taylor complexiﬁ- cation is reasonable and natural we get the equality of complex and real norms of operators between them.

An alternative approach to construct a reasonable norm on the complexiﬁcation space is to take into consideration all values of eiθz for 0 ≤ θ ≤ 2π. For example, Lindenstrauss and Tzafriri ([35], I, p. 81) used the following reasonable norm 1/2 − 2 2 x + iy LT =sup x cos θ y sin θ X + x cos θ + y sin θ X . 0≤θ≤2π 2.3. COMPLEXIFICATION OF REAL BANACH LATTICES 11

This approach produces a plenty of other examples of reasonable norms which can be found in the papers of Kirwan [31]andMuñoz-Sarantopoulos-Tonge [46]. We want to mention here that not all complex Banach spaces can be produced as complexifications of real Banach spaces. The existence of such space was proved by Bourgain [8] using probabilistic arguments and the first explicit example was constructed by Kalton [25].

2.3. Complexiﬁcation of real Banach lattices Additional structure of real Banach lattices allows to introduce some special but natural procedure of complexiﬁcation of these spaces (cf. Schaefer [52]).

Definition 2. A real Banach lattice is a Banach space X over the real numbers endowed with a partial order such that any two elements x, y ∈ X have an inﬁmum (denoted by x ∧ y := inf{x, y})andsupremum(x ∨ y := sup{x, y}), the positive cone E+ := {x ∈ E : x ≥ 0} is closed under addition and multiplication by nonnegative real numbers and the partial order is connected to the norm on X by the condition

(2.3) y ∈ X and |x|≤|y|⇒x ∈ X and x≤y, where |x| = x ∨ (−x).

Now assume that we have complexiﬁed a real vector lattice X in a natural way, i.e. XC = X + iX and consider for z = x + iy, x, y ∈ X the supremum:

(2.4) |z| =sup|x cos θ + y sin θ| . 0≤θ≤2π

Obviously, this mapping satisﬁes the following properties of absolute value function

|z| =0ifandonlyifz =0

|αz| = |α||z| for all α ∈ C and z ∈ XC

|z1 + z2|≤|z1| + |z2| for all z1,z2 ∈ XC

If XC is a complex vector lattice and x is a lattice norm on X, then the absolute value function (2.4) suggests the extension of the given lattice norm to XC by 2 2 1/2 | | | | | | ∈ C z XC = z X = ( x + y ) for all z = x + iy X . X Clearly, with this notion of norm the complexified space will possess lattice property (2.3). As an example of Banach lattice consider the space of real valued functions C(K) defined on a compact set K.ItwasshownbyMuñoz-Sarantopoulos-Tonge in [46] that the lattice complexification of this space coincides with the Taylor complexification. Thus, Proposition 2 implies the equality of the norms of any bounded linear operator between C(K) spaces and its complexification. Moreover, ∞ since LC space is isometrically isomorphic to CC(K) and since this isomorphism maps real functions into real functions, then the equality of real and complex norms holds even for operators between L∞ spaces. 12 2. COMPLEXIFICATION OF REAL SPACES AND OPERATORS

2.4. Complex norms of operators Applying any complexification procedure to real Banach spaces we extend bounded linear operators between these spaces to the complex bounded linear operators between their complexifications whose complex norms will be clearly not less than the real ones. In some cases we may even expect that complex operator norms remain the same as real. The problem of finding complexifications preserving operator norms was studied, for example, in [46]. Here we show how property (iii) of reasonable complexification procedure affects the relation between complex and ≤ real operator norms. It turns out that if the relations Re z X , Im z X z XC hold for any z = x + iy ∈ XC, then we easily get the following rough estimate of C T XC,YC / T X,Y . Proposition . · · 3 Let (X, X ) and (Y, Y ) be normed spaces with reason- C · C · ∈ able complexifications (X , XC ) and (Y , YC ) respectively. Then for any T L C ≤ C (X, Y ) we have T XC,YC 2 T X,Y , where T is a standard complexification of the operator T, i.e. TC = Tf + iT g.

Proof. ∈ C ≤ Recall that for any z X we have Re z X , Im z X z XC by the consequence of the conjugation-invariant property deﬁned in Deﬁnition 1. Then for z = x + iy ∈ XC T (x + iy) Tx+ iT y Tx + Ty Re z + Im z = ≤ ≤T ≤ 2 T . x + iy x + iy x + iy z

C ≤ Thus, T XC,YC 2 T X,Y . In general, constant 2 cannot be improved. Supply, for example, YC with maximal reasonable norm (2 x + iy T )andXC with minimal reasonable norm ( x + iy T ). ≤ Now assume that the condition Re z X , Im z X z XC does not hold and consider the following example of the complexification of a real Banach space showing that the relation between complex and real operator norms can be arbitrary large in this case. In particular, this relation is no longer necessarily bounded by constant 2 as we could intuitively expect. Example 2. Let 1 ≤ p<∞ and 0 <α<1/2. Consider space C2 as a natural complexification of R2 with the norm defined by: p p 1/p (z1,z2) =(α|z1 − iz2| +(1− α)|z1 + iz2| ) . As it was mentioned above this norm satisfies properties (i) and (ii) of Definition 2 1 but it is not conjugation-invariant. Clearly, for (x, y) ∈ R we have (x, y) = 2 2 x2 + y2. If we consider the√ operator T (x, y)=(x + y,x − y):R → R , then real norm of this operator is 2. On the other hand, taking√ (z1,z2)=(1,i) we 1/p 1/p obtain (1,i) =2α and TC(1,i) = (1 + i, 1 − i) =2 2(1 − α) . Hence, for 0 <α<1/2 1/p TC (1 − α) ≥ > 1. T α1/p We note, if α tends to zero, then (1 − α)1/p/α1/p tends to infinity. Thus, the relation between complex and real norms of this operator can be arbitrary large. CHAPTER THREE

Equality of Real and Complex Operator Norms

3.1. Operators between Lebesgue spaces In 1958 A. E. Taylor [56], referring to the paper of M. Riesz on convexity and bilinear forms [50], reproved Riesz’s statement about the equality of the real and complex norms for any bounded linear operator between finite-dimensional p q ≤ spaces ln and lm whenever p q and pointed out the importance of this result. Taylor’s proof used only standard calculus. Marcinkiewicz and Zygmund [44]used Gaussian variables to obtain the equality of real and complex norms of vector- valued operators between (quasi-)Banach Lp − Lp spaces. Later on A. Zygmund [64, p. 181] obtained the same result for the bounded linear operators from Lp[a, b] into Lp[a, b] space, relying his proof on a simple equality, showing that Lp[0, 1] 2 contains an isometric copy of l2. Using this nice idea of Zygmund, Verbicki˘ıand Sereda [59] in 1975 found a simple proof of the equality of real and complex norms p q for the operators between L (Ω1,μ)andL (Ω2,ν) spaces for 1 ≤ p ≤ q on any positive σ-finite measure spaces (Ω1,μ)and(Ω2,ν). In this chapter we present an alternative proof of Verbicki˘ı-Sereda theorem (cf. Holtz-Karow [23]), as well as find out some other possibilities when the norm of a real operator becomes equal to the norm of its complexification. As it was shown in the previous chapter such equality holds for bounded linear operators between Hilbert spaces, between C(K) spaces as well as between L∞ spaces. Here we obtain the same result for the positive operators between (quasi-)Banach lattices, for operators between some mixed (quasi-)norm Lebesgue spaces and between spaces of functions of bounded p-variation for some values of p. Finally, we give an example of an operator acting from an Orlicz space into the same Orlicz space for which the equality of complex and real norms does not hold. To obtain the most part of these results we will mainly look for the possibility to estimate the complex operator norm by the real one. In these estimates the fact that complex numbers possess symmetry of the unit ball will play a crucial role. By symmetry of the unit ball we mean that |a + ib| = |e−i2πt||a + ib| holds for all a, b, t ∈ R. Using Euler identity ei2πt =cos2πt + i sin 2πt we get |e−i2πt||a + ib| = |a cos 2πt + b sin 2πt + i(a sin 2πt − b cos 2πt)|. And since |z|≥|Re z| for any z ∈ C we have for all a, b, t ∈ R

|a + ib|≥|a cos 2πt + b sin 2πt|.

13 14 3. EQUALITY OF REAL AND COMPLEX OPERATOR NORMS

Taking, the p-power of the both sides of this inequality and integrating them in t over [0, 1] we get even equality which will be a very useful formula in our proofs. It is this formula was used by Zygmund when he showed the equality of complex and real norms for the operators between Lp[a, b] spaces. Lemma 1. For 0

Proof. We can assume that a2√+ b2 > 0, since otherwise we have equality. If we divide the both sides of (3.1) by a2 + b2 we obtain by 2π-periodicity of cosine that a b √ cos 2πt + √ sin 2πt = cos θ cos 2πt +sinθ sin 2πt p 2 2 2 2 L [0,1] a + b a + b Lp[0,1] 1 1/p | |p = cos(2πt + θ) Lp[0,1] = cos(2πt + θ) dt 0 1/p 1/p 2π+θ ds 2π ds = | cos s|p = | cos s|p θ 2π 0 2π 1 1/p | |p = cos 2πt dt = cos 2πt Lp[0,1] 0 and the proof is complete. Remark 2. Taking p = ∞ we get a useful particular case of this lemma, namely (a2 + b2)1/2 =sup{a cos 2πt + b sin 2πt :0≤ t ≤ 1}. Lemma 1 enables us to give a simple explanation why the norm of bounded linear operator T : Lp(μ) → Lp(ν) will not be changed when we go to the complex- iﬁed spaces. This proof was obtained by Zygmund [64, p. 181]. Thus, by Lemma 1 and the lattice structure of Lebesgue spaces we get | |2 | |2 1/2 dp TC(f + ig) p = dp ( Tf + Tg ) = Tfcos 2πt + Tgsin 2πt Lp[0,1] . p p Applying twice Fubini theorem and using boundedness of the operator T we obtain T (f cos 2πt + g sin 2πt) ≤T f cos 2πt + g sin 2πt p p p,p p p L [0,1] L [0,1] = T p,p f cos 2πt + g sin 2πt Lp[0,1] p = dp T p,p f + ig p . ≤ Hence, TC p,p T p,p and, since the reverse inequality is obviously true, we have TC p,p = T p,p . The constants dp will often appear in what it follows and that is why we em- phasize from the beginning some of their simple properties.

Lemma 2. Function dp is increasing in p for 0

Proof. Assume that 0

2 1 Remark 3. d =

Remark 5. In the case of quasi-Banach Lebesgue spaces it can happen that the operators T ∈L(Lp(μ),Lq(ν)) can be only trivial ones. For example, if T : Lp(μ) → Lq(ν) is a bounded linear operator, where μ is a non-atomic measure and 0 0 let T (l3,l3) be deﬁned by ε T (x1,x2,x3)=

(x1 − ε(x1 + x2 + x3),x2 − ε(x1 + x2 + x3),x3 − ε(x1 + x2 + x3)). Then for all couples (p, q) such as 0 0 such that ε ε T < TC. For the complete proof of this fact see Gasch-Maligranda [19]. ≤∞ Hence, if 0 1, that together with Corollary 1 implies Proposition . ≤ ≤∞ 4 γp,q =1if and only if 0

Theorem 2. If 0

Proof. The proof of this statement mainly relies on the integral Minkowski inequality. Let τ =min{r, s}, then τ τ τ τ τ dτ TC(f + ig) s,r = dτ Tf + iT g s r = Tfcos 2πt + Tgsin 2πt Lτ [0,1] s r 1/sτ 1 s/τ | |τ = Tfcos 2πt + Tgsin 2πt dt dμ1 Ω3 0 r 1/τ τ 1 | |τ = Tfcos 2πt + Tgsin 2πt dt 0 s/τ L r τ 1 1/τ τ ≤ |Tfcos 2πt + Tgsin 2πt| s/τ dt L 0 r 1/τ τ 1 = Tfcos 2πt + Tgsin 2πtτ dt s 0 r τ/r 1 r/τ τ = Tfcos 2πt + Tgsin 2πt s dt dμ2 Ω4 0 1 τ = Tfcos 2πt + Tgsin 2πt s dt r/τ 0 L 1 ≤ τ Tfcos 2πt + Tgsin 2πt s Lr/τ dt 0 1 τ = T (f cos 2πt + g sin 2πt) s,r dt 0 1 ≤ τ τ T f cos 2πt + g sin 2πt q,p dt. 0

Recall, τ =min{r, s}≥max{p, q}. Therefore, again by the integral Minkowski inequality we obtain

1 τ τ ≤ τ τ dτ TC(f + ig) s,r T f cos 2πt + g sin 2πt q,p dt 0 1 τ/p τ p = T f cos 2πt + g sin 2πt q dμ2 dt 0 Ω2 τ/p τ p = T f cos 2πt + g sin 2πt q dμ2 Ω2 Lτ/p[0,1] τ/p ≤T τ f cos 2πt + g sin 2πtp dμ q τ/p 2 Ω2 L [0,1] ⎛ ⎞ τ/p p/q τ ⎝ | |q ⎠ = T f cos 2πt + g sin 2πt dμ1 dμ2 Ω2 Ω1 Lτ/p[0,1] 3.2. OPERATORS BETWEEN SPACES OF BOUNDED p-VARIATION 19 ⎛ ⎞ p/τ τ/p 1 τ/q τ ⎝ | |q ⎠ = T f cos 2πt + g sin 2πt dμ1 dt dμ2 Ω2 0 Ω1

τ/p p/q τ | |q = T f cos 2πt + g sin 2πt dμ1 dμ2 Ω2 Ω1 Lτ/q[0,1] p/q τ/p ≤ τ | |q T f cos 2πt + g sin 2πt Lτ/q[0,1] dμ1 dμ2 Ω2 Ω1 p/q τ/p τ q = T f cos 2πt + g sin 2πt Lτ [0,1] dμ1 dμ2 Ω2 Ω1 τ τ τ = dτ T f + ig q,p .

Thus, TC≤T and since the reverse inequality is obvious the proof is complete.

Remark 8. By induction the same statement can be proved for the operators pn p1 qm q1 T ∈L(L [...[L ]...],L [...[L ]...], ),where0 < max{p1, ..., pn}≤min{q1, ..., qm}≤ ∞.

3.2. Operators between spaces of bounded p-variation Applying the same techniques as we used in the settings of Lebesgue spaces we can establish the equality of real and complex norms for operators between spaces of functions of bounded p-variation for some values of p. These spaces for p =2 were introduced by Wiener [62] when he investigated the convergence of Fourier series. The extension to 1 ≤ p ≤∞was done by L. C. Young and to all 0

Var∞(f)= sup |f(x)| . x∈[a,b] All functions (real or complex-valued) of bounded p-variation on [a, b] form a Ba- nach space (quasi-Banach space for 0

| | ∈ ∞ ∞ f BV∞ =sup f(x) , for f BV [a, b],where p = . x∈[a,b] 20 3. EQUALITY OF REAL AND COMPLEX OPERATOR NORMS

Clearly, the norm introduced in Deﬁnition 3 on the space BVp[a, b] is reasonable. Therefore, Proposition 3 proved for normed spaces with reasonable complexiﬁca- tions implies that TC≤2 T for any T ∈L(BVp[a, b]), where p ≥ 1. However, as in the case of Lebesgue spaces this estimate can be improved for some values of p and q. In this section we mainly consider the case when the complex norm of any operator T ∈L(BVp[a, b],BVq[a, b]) is the same as its real norm and show a general estimate of the relation TC / T valid even in quasi-Banach settings. Moreover, we obtain some cases of the equality of real and complex norms of operators T ∈ L(X, Y ) where one of the spaces X and Y is a space of functions of bounded p-variation and another one is Lebesgue or Hilbert space. Since we deal here with the quasi-Banach variant of the space BVp[a, b]for 0

Proposition 5. If 0

Proof. Assume that there exists some continuous function f ∈ BVp[a, b]which is not a constant. That means, that there exists x1 ∈ [a, b] such that f(a) = f(x1). Without loss of generality, let assume |f(a) − f(x1)| = 1 and put a =x0. ∈ − Now choose the smallest number x1/2 (x0,x1) such that f(x0) f(x1/2) = 1/2 whereof we have f(x1) − f(x1/2) ≥ 1/2. On the next step we choose two 2 numbers x1/22 ∈ (x0,x1/2)andx3/22 ∈ (x1/2,x1) such that for j =1, .., 2 we 2 have |f(xj−1) − f(xj )|≥1/2 . We continue this procedure in a similar way and each time choose 2k numbers, by one from each previously defined open segment, k k such that for j ∈ 1, ..., 2 we have |f(xj−1) − f(xj )|≥1/2 . Due to continuity of function f this construction is possible. Thus, taking the partition P as ordered rearrangement of xj we have k kp k(1−p) Varp(f) ≥ 2 /2 =2 which tends to infinity as k goes to infinity for 0

Corollary 2. If 0

Proof. Take any function from BVp[a, b]for0

Equality of the real and complex norms of operators T ∈L(BVp[a, b],BVq[a, b]) for either 0

Lemma 4. If either 0 sequence: n q/p q p q/p (3.4) Var (h) = lim |h(xk) − h(xk−1)| ≡ lim S , p n→∞ n→∞ n k=1 where a partition Pn : a = t1

Obviously, when 0

When p = ∞, then we clearly get sup Var∞(f cos 2πt+g sin 2πt)=Var∞(f +ig). 0≤t≤1 Considering the left inequality and taking into account that Varq(f) ≤ Varp(f) for all f ∈ BVp[a, b]and0

Corollary 3. If either 0

1/p p p ≤ |f(a)cos2πt + g(a)sin2πt| q/p + Var (f cos 2πt + g sin 2πt) L [0,1] p Lq/p[0,1] 1/p 1 p/q p| |p q = dq f(a)+ig(a) + Varp(f cos 2πt + g sin 2πt)dt 0 ≤ | |p p 1/p dq f(a)+ig(a) + Varp(f + ig) = dq f + ig BVp . The case when 0

Theorem 3. If either 0

Proof. Let f,g ∈ BVp[a, b]. From Lemma 4, Corollary 3 and boundedness of the operator T it follows that for q<∞ dq TC(f + ig) BVq = dq Tf + iT g BVq 1 1/q ≤ q T (f cos 2πt + g sin 2πt) BVq dt 0 1 1/q ≤ q T f cos 2πt + g sin 2πt BVp dt 0 ≤ dq T f + ig BVp .

Therefore, TC≤T and since the reverse inequality is obviously true, the theorem is proved. For q = ∞ the proof can be obtained by a simple modiﬁcation of these inequalities.

Remark 9. Particular case of Theorem 3 is the case of Banach spaces. Thus, if q ≥ 1, then for any T ∈L(BV1[a, b],BVq[a, b]) we have TC = T . Zygmund proved this statement for T ∈L(BV1[a, b]) by using random variables (see [64]).

Remark 10. The estimate TC / T ≤1/dq can now be easily obtained for all 0

The latter expression is clearly less than T sup f cos 2πt + g sin 2πt BVp . Using 0≤t≤1 again Corollary 3 we get

C ≤ dq T (f + ig) BVq T f + ig BVp .

If 0

Due to the similarity of inequalities (3.2) and (3.5) we can establish the following proposition.

q Proposition 6. For bounded linear operators T : BVp[a, b] → L (μ),where p 0

Proof. The proofs of these statements are just replications of the proofs presented above and based on Lemma 3, Corollary 3 and the standard complexiﬁcation of a Hilbert space.

Problem 1. Prove or disprove: if 1 T .

3.3. Operators between Banach lattices Besides the operators acting between Lebesgue spaces we are also interested in other examples of operators with equal real and complex norms. First, let consider positive operators acting between Banach lattices X and Y (see Chapter 2).

Definition 5. AlinearoperatorT : X → Y is said to be positive if for all f ∈ X with property f ≥ 0 we have Tf ≥ 0.

Definition 6. AlinearoperatorT : X → Y is said to be monotone if for all f,g ∈ X such that f ≥ g it follows Tf ≥ Tg.

The following lemma establishes the equivalence of these two deﬁnitions.

Lemma 5. An operator T is positive if and only if T is monotone.

Proof. Assume that the operator T is positive. Consider functions f and g such that f ≥ g.Thenf − g ≥ 0 and by the linearity and positivity of the operator T we have Tf − Tg = T (f − g) ≥ 0 or, equivalently, Tf ≥ Tg. The reverse implication is obvious, just take f ≥ 0=g and we get Tf ≥ 0.

Proposition 7. The equality TC = T holds for any positive operator T ∈ L(X, Y ), where X, Y are real (quasi-)Banach lattices.

Proof. Lattice complexiﬁcation of Y and Lemma 1 implies 1/2 2 2 C | | | | T (f + ig) YC = Tf + Tg = sup (Tfcos 2πt + Tgsin 2πt) . ≤ ≤ Y 0 t 1 Y 3.3. OPERATORS BETWEEN BANACH LATTICES 25

Hence, monotonicity of the operator T and lattice complexiﬁcation of X imply sup T (f cos 2πt + g sin 2πt) ≤ ≤ 0 t 1 Y ≤ T sup (f cos 2πt + g sin 2πt) ≤ ≤ 0 t 1 Y ≤T sup (f cos 2πt + g sin 2πt) ≤ ≤ 0 t 1 X | | = T f + ig X = T f + ig XC .

Hence, TC≤T and since the reverse inequality is obvious we have TC = T .

Now let consider arbitrary operators between (quasi-)Banach lattices. Minkowski inequality and its reverse which were the main tools in the proof of Theorem 2 and which appeared implicitly in the proof of Theorem 1 can be generalized to these spaces. Such generalizations need the notions of p-concavity and p-convexity. Namely, for 0

Theorem 4. Let X and Y be real (quasi-)Banach lattices with the Fatou property. If X is p-concave with constant 1 and Y is q-convex with constant 1 for 0

Maligranda [40]) and we get 1/q 1/q 1 1 T f cos 2πt + g sin 2πtq dt ≤T |f cos 2πt + g sin 2πt|qdt X 0 0 X = dq T f + ig X . Remark 11. If in Theorem 4 X is p-concave with constant 1 and Y is q-convex with constant 1 for 0

3.4. Operators between Orlicz spaces Up to now we have considered the case of the equality of real and complex norms of the operators from one (quasi-)Banach space into the same space. Recall, for example, operators T ∈L(Lp(μ)). Here, we will present an operator between the same two-dimensional Orlicz space for which real and complex norms will diﬀer. This Orlicz space will be constructed as an isometric copy of some symmetric space. Example 4. Consider a two-dimensional symmetric space X = R2 equipped for 1 ≤ p<2 with the norm 1 p p 1/p 2p (|x1| + |x2| ) +2 max{|x1|, |x2|} (3.6) (x1,x2) = 1 , x1,x2 ∈ R 1+22p and consider operator T : X → X given by the formula T (x, y)=(x + y,x − y). 3.4. OPERATORS BETWEEN ORLICZ SPACES 27

First, using Theorem 11, we ﬁnd the real norm of this operator: 1 p p 1/p 2p (|x1 + x2| + |x1 − x2| ) +2 max{|x1 + x2|, |x1 − x2|} T (x1,x2) = 1 1+22p 1 1 2p 1− p p p 1/p 2max{|x1|, |x2|} +2 2 (|x1| + |x2| ) ≤ 1 1+22p 1 p p 1/p 1 (|x | + |x | ) +22p max{|x |, |x |} 1 1− 2p 1 2 1 2 1− 2p ≤ 2 1 =2 (x1,x2) . 1+22p 1 1− 2p The equality holds for (x1,x2)=(1, 1) and therefore T =2 . On the other − 1 hand, the complex norm of this operator is strictly bigger than 21 2p since for 1 2 .

For p =1we take (z1,z2)=(1, 1+i) and get √ √ √ √ TC (z1,z2) / (z1,z2) = ( 2+1) 5+1 /(3 + 2) > 2.

Thus, TC / T > 1. Remark 12. For p =1this example was considered by Gasch-Maligranda [19]. We can even construct another example, taking 2 T . Proof. WeuseheretheresultofGrz¸a´slewicz [20]sayingthateverycompact symmetric convex subset of R2 with non-empty interior is a unit ball of some Orlicz ϕ space l2 . For simplicity we consider two-dimensional space X from Example 4 with p = 1. Clearly, the unit ball deﬁned by the norm (3.6) is a compact symmetric convex non-empty subset of R2 and therefore, due to Grz¸a´slewicz result, it is a unit ball of some Orlicz space. LetustakeOrliczspacelϕ,where 2 √u , 0 ≤ u ≤ √1 (3.7) ϕ(u)= 2 2 . u(1 + √1 ) − √1 , √1 ≤ u 2 2 2 It is simple to check that Luxemburg norm on lϕ which is given by 2 x x (x ,x ) =inf{k>0:ϕ 1 + ϕ 2 ≤ 1} 1 2 ϕ k k coincides with the original norm (3.6) on X. Indeed, without loss of generality let ﬁx 0 ≤ x ≤ x . Then, (x ,x ) = x + x√1 .Tocalculate(x ,x ) =inf{k> 1 2 1 2 2 1+ 2 1 2 ϕ x1 x2 ≤ } 0:ϕ( k )+ϕ( k ) 1 we consider the following three cases obtained naturally from the representation of ϕ,namely

(x1,x2)ϕ =min{inf A, inf B,inf C}, 28 3. EQUALITY OF REAL AND COMPLEX OPERATOR NORMS where √ x x x + x 1 √ A = {k>0:k ≤ 2x and ϕ( 1 )+ϕ( 2 )=( 1 2 )(1 + √ ) − 2 ≤ 1}. 1 k k k 2 √ √ √ x1 x2 x1 x2 1+ 2 1 B = {k>0: 2x1 ≤ k ≤ 2x2 and ϕ( )+ϕ( )=√ + √ − √ ≤ 1}. k k 2k k 2 2 √ x1 x2 x1 + x2 C = {k>0:k ≥ 2x2 and ϕ( )+ϕ( )= √ ≤ 1}. k k k 2 √ √ √ The set A consists only of 2x since x1+x2 ≤ 1+ √2 = 2 and to satisfy x1√+x2 ≤ √ 1 k 1+1/ 2 √ 2 k ≤ 2x ≤ x1√+x2 we should have x = x and therefore k = 2x .Since √1 2 1 √ 2 √ 1 inf C ≥ 2x and in the set B we have 2x ≤ k ≤ 2x and k ≥ x + x√1 , 2 √ √ 1 2 √ 2 1+ 2 and also since 2x ≤ x + x√1 ≤ 2x and x + x√1 = 2x it follows that 1 2 1+ 2 2 1 1+ 2 1 (x ,x ) =infB = x + x√1 , and the equality (x ,x ) = (x ,x ) is 1 2 ϕ 2 1+ 2 1 2 ϕ 1 2 proved. Since the symmetric space X has the property TC / T > 1forT ∈L(X, X), ϕ then Orlicz space l2 which is isometric to X will inherit this property. So, for the − ϕ bounded linear operator T (x, y)=(x + y,x y) between Orlicz spaces l2 with ϕ defined by (3.7) real and complex norms are different. Remark 13. Using the construction of Example 4 with 1 0 and some positive σ-finite measure μ. It is interesting to note that for any symmetric space X on [0, 1] and any ∈ R θ we have cos (2πt + θ) X[0,1] = cos 2πt X[0,1] due to equimeasurability of the functions cos (2πt + θ)andcos2πθ. Hence, the following lemma similar to Lemma3works. Lemma 6. If X is a symmetric space on [0, 1] and a, b ∈ R,then 2 2 a cos 2πt + b sin 2πt X[0,1] = dX a + b , where dX = cos 2πt X[0,1] . Remark 14. Analyzing the proof of Theorem 1 we note that we applied twice Fubini theorem, namely each time we invoked Lemma 3. Applying twice Fubini theorem is essentially the same as changing the order of taking the norms in a mixed norm. The latter is possible in the case of Lp[Lp] space and by Kolmogorov- Nagumo theorem it is the only possibility among all mixed norm spaces composed of symmetric spaces (see Boccuto-Bukhvalov-Sambucini [7]). CHAPTER FOUR

Basic Properties of Constant γX,Y

This chapter is devoted to the discussion of some basic properties of the constant γ(X, Y ), where spaces X and Y can be Lebesgue spaces, mixed norm Lebesgue spaces or spaces of functions of bounded p-variation. We also establish here that for quasi-Banach variants of these spaces the relation between complex and real norms of bounded linear operators between them is bounded by 2 as it is in the Banach case. Theorem 6 (cf. Verbicki˘ı[60]). (i) If 0

TCp,q ≤ dp/dqT p,q.

dp dq Moreover, if 1 ≤ q ≤ p ≤∞,thenTC ≤ min{ , }T . p,q dq dp p,q (ii) If 0

Thus, if 0

29 30 4. BASIC PROPERTIES OF CONSTANT γX,Y where a(f)=2 1 f(t)cos2πt dt, b(f)=2 1 f(t)sin2πt dt and f ∈ Lp(0, 1). Ap- 0 0 2 2 1/2 plying Lemma 1 twice we get Ffq = dq a (f)+b (f) = dq/d2Ff2.Con- sequently,

(4.2) F p,q = dq/d2F p,2. ∗ ∗ By duality we obtain F p,2 = F 2,p and F is deﬁned by F ∗ϕ(x)=a(ϕ)cos2πx + b(ϕ)sin2πx, 1 1 where a(ϕ)=2 0 ϕ(t)cos2πt dt and b(ϕ)=2 0 ϕ(t)sin2πt dt. Applying Lemma 1gives ∗ 2 2 1/2 F ϕp = dp [a (ϕ)+b (ϕ)] . ∗ From Parseval identity it follows that the norm F 2,p is attained on the function ϕ(x)=a(ϕ)cos2πx + b(ϕ)sin2πx. The norm of this function in L2(0, 1) can be derived by using Lemma 1 again 1 2 1/2 2 2 1/2 ϕ2 =( |a(ϕ)cos2πt + b(ϕ)sin2πt| dt) = d2[a (ϕ)+b (ϕ)] . 0 ∗ ∗ Thus, F p,2 = F 2,p = F ϕp /ϕ2 = dp /d2. Substituting this expression into the equality (4.2) we obtain 2 (4.3) F p,q = dp dq/d2.

Let FC be a complexiﬁcation of operator F , then the expression (4.1) gives 2πix 2πix FC(e )=F (cos 2πx)+iF (sin 2πx)=cos2πx + i sin 2πx = e , that implies 2πix 2πix 2πix 2πix FCp,q ≥FC(e )q/e p = e q/e p =1.Now,sinceFCp,q ≥ 2 ≥ 1=d2/(dp dq) F p,q by the equality (4.3) and FC p,q F p,q,wehave 2 ≥ { } FC p,q max 1,d2/(dp dq) F p,q.

Corollary . ≥ ≤ 4 If p 2,thenγp,2 = d2/dp and if 0

Proof. ≤ { }≤ By Theorem 6(i) we have γp,2 min dp/d2,d2/dp d2/dp as well ≤ as by the preceding remark γ2,q d2/dq. Besides, Theorem 6(ii) states that there exist operators F and G with the properties FC2,p /F 2,p ≥ d2/dp and GC2,q/G2,q ≥ d2/dq. These facts yield the identities of this corollary. Corollary . ≤ ≤∞ ≤ π 5 If 1 p, q , then γp,q 2 . Proof. By Theorem 6 and Lemma 2 we have

≤ {dp dq }≤d∞ π γp,q min , = . dq dp d1 2 Lemma 7. If 1 ≤ q

Consider the case when 1/p+1/q ≥ 1. This inequality obviously implies 1/q ≤ 1/p ≥ q−p q and therefore q p>p.Takingθ = q−q p we can easily check that 0 <θ<1as 1 1−θ θ well as that θ satisﬁes the equality p = q + q . Now we can apply Riesz-Thorin interpolation theorem (see Chapter 7) and get ≤ 1−θ θ F p,p F q,q F q,q = F q,q, that due to relation (4.4) is equivalent to the inequality dq /dp ≥ dp/dq.Thecase when 1/p +1/q < 1 can be considered analogously.

We note that by the remark which follows Lemma 2 we obtain √ p+1 1/p q+2 1/q 1/q−1/p Γ( 2 ) Γ( 2 ) dp/dq = π p+2 q+1 . Γ( 2 ) Γ( 2 ) p,n p Denote by T the subspace of L [0, 2π] consisting of all trigonometric poly- n nomials tn(x)=a0 + k=1(ak cos kx+ bk sin kx). Using Bernstein lemma Verbicki˘ı [60] found the best constant γ(T p,n, T q,m) for all 1 ≤ p, q ≤∞and the extension of this result to 0

an cos nx + bn sin nxp ≤tnp p,n for any trigonometric polynomial tn ∈T . Proof. (cf. Achieser [1, pp. 12-13]). First we assume that 1

(4.5) pn(x + π/n)=−pn(x) and consequently

(4.6) pn(x +2π/n)=pn(x).

The latter expression proves that pn(x)hastheperiod2π/n and hence

an−1 = bn−1 = ... = a1 = b1 =0, while the relation (4.5) gives that

a0 =0. 32 4. BASIC PROPERTIES OF CONSTANT γX,Y

Thus, for p>1 n an cos nx + bn sin nxp ≤a0 + ak cos kx + bk sin kxp. k=1 By continuity principle we also have that Bernstein lemma is true for p =1and p = ∞. Proposition 8. For any m, n ∈ N the following relation holds: 1, if 1 ≤ p ≤ q ≤∞ γ(T p,n, T q,m)= dp/dq, if 0

Ftn(x)=an cos mx + bn sin mx.

As usual, denote by FC the complexiﬁcation of F ,then inx imx FCe = F (cos nx)+iF (sin nx)=cosmx + i sin mx = e . iθ iθ 1/p The evident equality |e | =1foranyθ ∈ R implies e p =(2π) . Hence, iθ iθ 1/q−1/p (4.7) FCp,q ≥FCe q/e p =(2π) .

Furthermore, by Bernstein lemma we have tnp ≥an cos nx + bn sin nxp and 1 1 p 2 2 a cos nx + b sin nxp =(2π) dp(a + b ) 2 is easily derived from Lemma 1. There- 2 2 fore, for an + bn > 0wehave Ftn q an cos mx + bn sin mx q 1/q−1/p ≤ =(2π) dq/dp tnp an cos nx + bn sin nxp p,n and taking the supremum of the both sides of this inequality over tn ∈T with tnp =1weget 1/q−1/p (4.8) F p,q ≤ (2π) dq/dp.

The combination of relations (4.7) and (4.8) gives FCp,q/F p,q ≥ dp/dq.Atthe same time, Theorem 6(i) states that TCp,q/T p,q ≤ dp/dq for 0 2=γ∞,1.

In the following theorem we consider the constants γX,Y ,whereX and Y are mixed (quasi-)norm Lebesgue spaces. Theorem 7. If 0 < min{r, s}≤max{p, q}≤∞, then for arbitrary operator T ∈L(Lp[Lq],Lr[Ls]) we have

TC≤dmax{p,q}/dmin{r,s}T .

dmax{p,q} dmax{r,s} Moreover, if 1 ≤ min{r, s}≤max{p, q}≤∞,thenTC≤min{ , }T . dmin{r,s} dmin{p,q} 4. BASIC PROPERTIES OF CONSTANT γX,Y 33

Proof. Set τ =min{r, s}. Repeating the first step of the proof of Theorem 2 we obtain 1 τ τ ≤ τ dτ TC(f + ig) s,r T (f cos 2πt + g sin 2πt) s,r dt. 0 Now, using boundedness of the operator T ,Hölder inequality, Fubini theorem and assuming that τ ≤ p we have 1 τ τ ≤ τ τ dτ TC(f + ig) s,r T f cos 2πt + g sin 2πt q,p dt 0 1 τ/p τ p = T f cos 2πt + g sin 2πt q dμ2 dt 0 Ω2 1 τ/p ≤ τ p T f cos 2πt + g sin 2πt q dμ2dt 0 Ω2 1 τ/p ≤ τ p T f cos 2πt + g sin 2πt q dtdμ2 Ω2 0 τ/p 1 p/q τ | |q = T f cos 2πt + g sin 2πt dμ1 dtdμ2 . Ω2 0 Ω1 If p ≥ q,then τ/p p/q τ τ ≤ τ | |q dτ TC(f + ig) s,r T f cos 2πt + g sin 2πt dμ1 dμ2 Ω2 Ω1 Lp/q[0,1] p/q τ/p ≤ τ | |q T f cos 2πt + g sin 2πt Lp/q[0,1] dμ1 dμ2 Ω2 Ω1 p/q τ/p τ q = T f cos 2πt + g sin 2πt Lp[0,1] dμ1 dμ2 Ω2 Ω1 τ τ τ = dp T f + ig q,p . If conversely, q ≥ p, then by the Hölder inequality and Fubini theorem τ/p 1 p/q τ τ ≤ τ | |q dτ TC(f + ig) s,r T f cos 2πt + g sin 2πt dμ1 dtdμ2 Ω2 0 Ω1 τ/p 1 p/q ≤ τ | |q T f cos 2πt + g sin 2πt dμ1dt dμ2 Ω2 0 Ω1 τ/p 1 p/q τ | |q = T f cos 2πt + g sin 2πt dtdμ1 dμ2 Ω2 Ω1 0 τ τ τ = dq T f + ig q,p . If q ≥ τ ≥ p, then by the integral Minkowski and Hölder inequalities we obtain 1 τ τ ≤ τ τ dτ TC(f + ig) s,r T f cos 2πt + g sin 2πt q,p dt 0 34 4. BASIC PROPERTIES OF CONSTANT γX,Y 1 τ/p τ p = T f cos 2πt + g sin 2πt q dμ2 dt 0 Ω2 τ/p τ p = T f cos 2πt + g sin 2πt q dμ2 Ω2 Lτ/p[0,1] τ/p ≤T τ f cos 2πt + g sin 2πtp dμ q τ/p 2 Ω2 L [0,1] ⎛ ⎞ p/τ τ/p 1 τ/q ≤ τ ⎝ | |q ⎠ T f cos 2πt + g sin 2πt dμ1 dt dμ2 Ω2 0 Ω1

τ/p 1 p/q ≤ τ | |q T f cos 2πt + g sin 2πt dμ1dt dμ2 Ω2 0 Ω1 τ τ τ = dq T f + ig q,p .

If 1 ≤ min{r, s}≤max{p, q}≤∞, then by duality we also obtain TC/T ≤ dmax{r,s}/dmin{p,q}.

Remark . ≤ 16 By induction we get TC dmax{p1,..,pn}/dmin{q1,...,qm} T for pn p1 qm q1 any T ∈L(L [...[L ]...],L [...[L ]...]),where0 < min{q1, ..., qm}≤max{p1, ..., pn}≤ ∞.

Problem 4. Deﬁne the conditions on 0

p q r s γ(L [L ],L [L ]) = min{dmax{p,q}/dmin{r,s},dmin{r,s}/dmax{p,q}}. In the next theorem we summarize the known and just obtained properties of the constant γX,Y . √ Theorem . 8 (i) (Krivine [34]) γ∞,1 = 2. (ii) (Krivine [34], Gasch-Maligranda [19]) γp,q is increasing in p and decreas- ≤ ≤∞ ≤ ing in q for 1 p, q and γp,q γ∞,1. ≤ ≤∞ (iii) If 1 p, q ,thenγp,q = γq,p . ≤ ≤∞ (iv) γp,q =1if and only if 0

(x) If 0 triangle inequality in the proof of Proposition 3 we get for 0

CHAPTER FIVE

Clarkson Inequality in the Real and the Complex Case

In this chapter we consider the so called generalized Clarkson inequality (|a + b|q + |a − b|q)1/q ≤ C (|a|p + |b|p)1/p , where 0

5.1. The best constant in the complex case We will look ﬁrst at a generalized complex Clarkson inequality. The result is, in fact, known but our formulation of Theorem 10 and its proof will help to understand better the statement and the proof of the new result in Theorem 11 as well as the applications of both theorems presented in the third section of this chapter. Theorem 10. Let 0

1/p Clarkson [11]provedthatCp,p (C)=2 for 1 ≤ p ≤ 2, where p is the conjugate exponent to the number p,thatis,1/p +1/p = 1. Later on the best constants Cp,q(C) for the remaining pairs of p and q such that 0

37 38 5. CLARKSON INEQUALITY IN THE REAL AND THE COMPLEX CASE

Proof. We carry out the proof in three steps. First step. For 1 ≤ p ≤ 2the following classical complex Clarkson inequality (cf. [11]) holds 1/p (5.2) |a + b|p + |a − b|p ≤ 21/p (|a|p + |b|p)1/p ∀a, b ∈ C.

1 → ∞ Really, we can calculate the norm of the operator T in two easy cases: T : l2 l2 | | | − | 2 → 2 has the norm T 1,∞ =supmax ( x + y , x y )=1andT : l2 l2 has |x|+|y|=1 √ 2 2 1/2 the norm T 2,2 =sup(|x + y| + |x − y| ) = 2. Then, by the complex |x|2+|y|2=1 p C → q C p C ≤ ≤ Riesz-Thorin interpolation theorem, T : l2( ) l2( )=l2 ( )(1 p 2) is bounded with the norm 1−2/p 2/p 1/p ≤ T p,p T 1,∞ T 2,2 =2 , since p and q are defined by 1 1 − θ θ 1 1 − θ θ = + , = + and 0 ≤ θ ≤ 1. p 1 2 q ∞ 2 p p 1/p p Second step. For fixed a, b ∈ C let Ap =(|a| + |b| ) and Bp =(|a + b| + p 1/p −1/p −1/p |a − b| ) .ThenAp,Bp are decreasing and 2 Ap, 2 Bp are increasing in p>0. The proof of these statements is standard. Third step. Combine the first two steps by considering special four cases for p and q: 1/q−1/2 1/q 1/q−1/p+1/2 I. 0

Now consider the function gr(θ)foraﬁxedr ∈ [0, 1] given by iθ p iθ p gr(θ)=|1+re | + |1 − re | =(1+r2 +2r cos θ)p /2 +(1+r2 − 2r cos θ)p /2. Its derivative is − 2 p/2−1 − 2 − p/2−1 gr(θ)= p r sin θ (1 + r +2r cos θ) (1 + r 2r cos θ) 5.2. THE BEST CONSTANT IN THE REAL CASE 39

kπ and gr(θ) = 0 if and only if θ = 2 for k =0, 1, 2, 3, 4. Moreover, gr is decreasing π 3π π 3π on intervals [0, 2 ], [π, 2 ] and increasing on [ 2 ,π], [ 2 , 2π]. Therefore,

p p max gr(θ)=gr(0) = gr(π)=(1+r) +(1− r) θ∈[0,2π] and it only remains for us to prove (5.4) for θ =0,thatis

(5.5) (1 + x)p +(1− x)p ≤ 2(1 + xp)p −1 for all 0 ≤ x ≤ 1and1

f(α, x)=(1+α1−p x)(1 + αx)p −1 +(1− α1−p x)(1 − αx)p −1 of real variables 0 ≤ α, x ≤ 1. Obviously,

f(1,x)=(1+x)p +(1− x)p ,f(xp−1,x)=2(1+xp)p −1 and − − −p p −2 − − p−2 fα(α, x)=(p 1)x(1 α ) g(α, x), where g(α, x)=(1+αx) (1 αx) . ≥ ≤ Clearly, for p > 2wehaveg(α, x) 0, that in turn implies fα(α, x) 0, since 1−α−p ≤ 0. Thus, the function f(α, x) is decreasing in α.Notingthatxp−1 ≤ 1we have f(1,x) ≤ f(xp−1,x)thatisthesameas(1+x)p +(1−x)p ≤ 2(1+xp)p −1.

Remark 17. The proof of the classical complex Clarkson inequality by using the complex Riesz-Thorin interpolation theorem gives possibility to do some generalizations of the classical complex Clarkson inequality to the higher dimensions (see [63], [41], [26], [27] and [28]).

5.2. The best constant in the real case Now, we consider a generalized real Clarkson inequality:

(5.6) (|a + b|q + |a − b|q)1/q ≤ C (|a|p + |b|p)1/p ∀a, b ∈ R.

The problem is how to compute the best constant C = Cp,q(R) in this inequality for all 0

Moreover, if either p ≤ 2 or q ≥ 2 or p = ∞ or 0

Proof. If either 0

Figure 1. The best Figure 2. The best constants in the generalized constants in the generalized complex Clarkson inequality. real Clarkson inequality.

Remark 18. Let 1 ≤ p, q ≤∞.Ifdim Lp(μ) ≥ 3 and dim Lq(ν) ≥ 3, then the norm of any bounded linear operator T : Lp(μ) → Lq(ν) acting between real spaces p q and the norm of its natural complexiﬁcation TC = T + iT : LC(μ) → LC(ν) acting between complex spaces are the same T p,q = TCp,q if and only if p ≤ q (see [56], p p q q [19] and [39]). In two-dimensional case, i.e. when L (μ)=l2 and L (ν)=l2, we have that T p,q = TCp,q if and only if either 1 ≤ p ≤ 2 or 2 ≤ q ≤∞(see [59], [19] and [39]). 5.2. THE BEST CONSTANT IN THE REAL CASE 41

Theorem 12. For all 1

Now, in order to prove that h(x) > 0 for all x ∈ (0, 1) and 2

Hence, φk(x) >φk(1) = 0. Thus, all functions φk(x) are positive which clearly implies that φ(x) > 0 and therefore h(x) > 0 for all required x and p, i.e. the function h(x) is strictly increasing on (0, 1). It remains to show that h((0, 1)) = (0, 1). For p ∈ (2, ∞)andx ∈ (0, 1) we have that xp−1

Therefore, both u(x)andv(x) are decreasing on (0, 1), u(0) = 0 and v(0) = −2(q − 1). Moreover, u(0) = v(0) = 1. Hence, u(x) − v(x) > 0 which means that F (x) is increasing at this neighborhood. Thus, x0 is the point of maximum of F (x) for p ∈ (2, ∞)andq ∈ (1, 2). 1−1/p 1/q 1/q−1/p+1/2 It remains for us to show that max{2 , 2 }

1/q−1/p+1/2 and thus Bp,q = F (x0) < 2 . Remark 19. For p>2 Thorin in [58] proved a stronger result, namely that the function F (x, p, p) deﬁned in Theorem 12 is logarithmically concave in 1/p. Theorem 12 only shows the existence of a point inside the interval (0, 1) at which the function F (x) given by (5.8) has its maximum. But it is even more interesting to ﬁnd this point and as a consequence to obtain the best constant in the generalized real Clarkson inequality for 1

If we denote the root of (5.13) by x0,thenmax{F (x):x ∈ (0, 1)} = F (x0)and thus the best constant in the generalized real Clarkson inequality for 2

3 1/3 After substitution this expression into (5.14) we get C 3 (R)=(1+x )(1+x ) ≈ 3, 2 0 0 5/6 1.622. AtthesametimewehaveC 3 (C)=2 ≈ 1.781 and so C 3 (C)/C 3 (R) ≈ 3, 2 3, 2 3, 2 1.098. 44 5. CLARKSON INEQUALITY IN THE REAL AND THE COMPLEX CASE

The constant C 4 (R)(p = 4) is connected to the real root of the equation 4, 3 x4 + x3 + x − 1=(x2 +1)(x2 + x − 1) = 0, √ √ which is x0 =( 5 − 1)/2 ∈ (0, 1) and therefore C 4 (R)= 3 ≈ 1.732. We also 4, 3 √ have that C 4 (C) = 2 and thus C 4 (C)/C 4 (R)=2/ 3 ≈ 1.154. 4, 3 4, 3 4, 3 When p = 8 the real root of the equation x8 + x7 + x − 1=(x2 +1)(x3 − x +1)(x3 + x2 − 1) = 0. can be found from x3 + x2 − 1 = 0 and is equal to √ √ 3 3 x0 =1/3 25 + 3 69 /2+ 25 − 3 69 /2 − 1 ≈ 0.755.

5/4 Therefore, C8,8/7(R) ≈ 1.892 and since C8,8/7(C)=2 ≈ 2.378 it follows that C8,8/7(C)/C8,8/7(R) ≈ 1.256. For the remaining values of p ∈ (2, ∞) we could not calculate exactly the constants Cp,p (R) but we noticed that (5.13) can be solved for p : 1 − x (5.15) p(x)=1+log . x 1+x √ Now, taking arbitrary value of x ∈ [ 2 − 1, 1) we ﬁnd the corresponding value p and therefore by (5.14) we can calculate the constant Cp,p’ (R) for this particular p. The left end of the interval is determined by putting p = 2 in (5.15) and solving this√ for x (limit case). It is also worth to note that the function p(x)isincreasing on [ 2 − 1, 1). Indeed, its derivative 1−x 2 1 1−x ln − 2 ln x − ln p(x)= 1+x = 1−x x 1+x ≥ 0. ln x (ln x)2

≈ R log63/2 ≈ For example, if we take x =1/2, then p =1+log2 3 2.584, Cp,p ( )=7 1/2+log 3/2 1.553, Cp,p (C)=2 6 ≈ 1.654 and thus Cp,p (C)/Cp,p (R) ≈ 1.065. If x0 = ≈ R ≈ C ≈ 9/10, then p =1+log9/10 1/19 28.9463, Cp,p ( ) 1.9836, Cp,p ( ) 2.6962 and C R ≈ ≈ Cp,p ( )/Cp,p ( ) 1.3592. For x0 =99/100 we have p =1+log99/100 1/199 R ≈ C ≈ C R ≈ 527.6794, Cp,p ( ) 1.9999, Cp,p ( ) 2.8210 and Cp,p√ ( )/Cp,p ( ) 1.4106. Observe that Cp,p (C)/Cp,p (R) → C∞,1(C)/C∞,1(R)= 2asp approaches ∞ (see Krivine [34]).

5.3. Some applications of the Clarkson inequality In this section we show how the real and complex generalized Clarkson inequalities can be used in calculation of the norm of a certain operator as well as in the proof of the (p, q)-Clarkson inequalities in Lebesgue spaces, mixed norm spaces and in normed spaces. For 1 ≤ p, q ≤∞consider a linear operator T : Lp[a, b] → Lq[a, b]givenby b−a ∈ a+b x(t)+x(t + 2 ),t a, 2 Tx(t)= − b−a − ∈ a+b x(t 2 ) x(t),t 2 ,b . As an application of Theorems 10 and 11 we calculate the norm of this operator acting between complex and real spaces. 5.3. SOME APPLICATIONS OF THE CLARKSON INEQUALITY 45

Theorem 13. Let 1 ≤ p, q ≤∞. Then the operator T : Lp[a, b] → Lq[a, b] is bounded if and only if p ≥ q. Moreover, p−q b − a pq (5.16) T = C ,whereC = C (R) or C (C) p,q 2 p,q p,q p,q p,q depending if the spaces are real or complex. Proof. First, by Theorems 10, 11 (substituting a = x + y and b = x − y for x, y ∈ R or x, y ∈ C)

1/q (a+b)/2 − b − b a q b a q Txq = |x(t)+x(t + )| dt + |x(t − ) − x(t)| dt a 2 (a+b)/2 2 1/q (a+b)/2 b − a (a+b)/2 b − a = |x(t)+x(t + )|qdt + |x(s) − x(s + )|qds a 2 a 2 1/q (a+b)/2 b − a b − a = |x(t)+x(t + )|q + |x(t) − x(t + )|q dt a 2 2 1/q (a+b)/2 b − a ≥ min{21/q, 21/q } |x(t)|q + |x(t + )|q dt a 2 1/q (a+b)/2 b b − a =min{21/q, 21/q } |x(t)|qdt + |x(t + )|qdt a (a+b)/2 2 1/q 1/q =min{2 , 2 }xq. Thus, in the case p

1/q (a+b)/2 − b − b a q b a q Txq = |x(t)+x(t + )| dt + |x(t − ) − x(t)| dt a 2 (a+b)/2 2 1/q (a+b)/2 − q/p p b a p ≤ Cp,q |x(t)| + |x(t + )| dt a 2 1/p p−q (a+b)/2 − (a+b)/2 pq p b a p ≤ Cp,q (|x(t)| + |x(t + )| )dt dt a 2 a p−q b − a pq = C x , p,q 2 p where Cp,q = Cp,q(R) in the real case and Cp,q = Cp,q(C) in the complex case. To verify that equality may occur in (5.16) we can take a pair (u, v)ofnumbers from Theorem 10 and Theorem 12 (in the case 1

x(t)=uχ[a,(a+b)/2)(t)+vχ[(a+b)/2,b](t). The main application of the classical complex Clarkson inequality (5.2) was for Clarkson [11]toprovethatfor1 0 such that for all x, y ∈ X with x = y =1andx − y≥ε it follows that (x + y) /2 < 1 − δ. The function δX (ε):[0, 2] → [0, 1] deﬁned by the formula x + y δX (ε)=inf{1 − : x, y ∈ SX , x − y≥ε} 2 is called the modulus of uniform convexity of X. Clearly, a normed space X is uniformly convex if δX (ε) > 0 for all 0 <ε≤ 2. Using inequality (5.2) Clarkson proved two inequalities, one for 1

1/r 1/r r − r ≤ 1/r r r ∈ p x + y p + x y p 2 x p + y p for all x, y L .

− ≥ ∈ ≤ − r 1/r Thus, if x p = y p =1and x y ε, ε [0, 2], then x + y p 2[1 (ε/2) ] and so r1/r r ε 1 ε (5.17) δ p (ε) ≥ 1 − 1 − ≥ , where r =max{p, p } . L 2 r 2 For 2 ≤ p<∞ estimate (5.17) is sharp. For 1 8 ε was found by Hanner [21] and Kadec [24]. Later on Boas [6] obtained, by the Riesz convexity theorem, the following result: ≤ ≤ q − q 1/q ≤ 1/p p p 1/p ∈ if 1

Some results on a generalization of (5.18) to the mixed norm spaces were obtained by Boas [6], Koskela [30] and Sobolev [54] and this work was continued by Kato-Miyazaki [26] and Kato-Miyazaki-Takahashi [27], [28]. In contrast with the proofs presented in [26], [27]and[29], where the interpolation techniques were used, our proof of the (p, q)-Clarkson inequality in the mixed norm spaces will only rely on the Jensen, Minkowski and (p, q)-Clarkson inequalities in Lr(μ) spaces. We will also complete the chain of generalizations of the (p, q)-Clarkson inequality to the mixed norm spaces. Let us recall that the mixed norm spaces Ls [Lr] (cf. Benedek-Panzone [2]) are classes of measurable functions on Ω1 × Ω2 generated by the quasi-norms (norms when r, s ≥ 1)

1/s s/r | |r x r,s = x r s = x(t1,t2) dμ1 dμ2 , where r, s > 0. Ω2 Ω1

Theorem 14. If 0

1/q 1/p q − q ≤ 1/q−1/p+1/t p p (5.20) x + y r,s + x y r,s 2 x r,s + y r,s

for all x, y ∈ Ls[Lr],where t =min{p, q,r,r,s,s} and the conjugate exponent r ≥ is omitted if the number is not bigger than one. Moreover, if dim L (μ1) 2 and s ≥ 1/q−1/p+1/t dim L (μ2) 2, then the constant 2 is sharp.

Proof. We prove the inequality for real positive numbers p, q, r and s.Simple modiﬁcation of this proof establishes that inequality (5.20) also holds when some of these numbers are equal to inﬁnity. Case 1: If s ≤ q, then by applying the reverse Minkowski inequality we get

1 q q q 1 s s q s s q − q q q − q q x + y r,s + x y r,s = x + y r dμ2 + x y r dμ2 Ω2 Ω2 1/s ≤ q − q s/q (5.21) ( x + y r + x y r) dμ2 . Ω2

Further, if p ≤ s, then by sequential use of the (p, q)-Clarkson and Minkowski inequalities it follows 1/q 1/s q − q ≤ 1/q−1/p+1/t p p s/p x + y r,s + x y r,s 2 ( x r + y r ) dμ2 Ω2 p/s p/s 1/p ≤ 1/q−1/p+1/t s s 2 x r dμ + y r dμ Ω2 Ω2 1/p 1/q−1/p+1/t p p =2 x r,s + y r,s , where t =min{p, q,r,r} . Since s ≤ q (or q ≤ s)andp ≤ s we can write t =min{p, q,r,r,s,s} . 48 5. CLARKSON INEQUALITY IN THE REAL AND THE COMPLEX CASE

If we instead have p ≥ s and apply the (s, q)-Clarkson inequality to (5.21) we get 1/q 1/s q − q ≤ 1/q−1/s+1/t s s x + y r,s + x y r,s 2 ( x r + y r) dμ2 Ω2 1/s 1/q−1/s+1/t s s =2 x r,s + y r,s 1/p ≤ 1/q−1/p+1/t p p 2 x r,s + y r,s , where t =min{s, q,r,r} =min{p, q,r,r,s,s} , since s ≤ q (q ≤ s)ands ≤ p. Case 2: If s ≥ q, then 1/q 1/s q − q ≤ 1/q−1/s s − s x + y r,s + x y r,s 2 x + y r,s + x y r,s 1/s 1/q−1/s s − s (5.22) =2 ( x + y r + x y r) dμ2 . Ω2 Moreover, if s ≤ p then by the (s, s)-Clarkson inequality we have 1/q 1/s q − q ≤ 1/q−1/s+1/t s s x + y r,s + x y r,s 2 ( x r + y r) dμ2 Ω2 1/s 1/q−1/s+1/t s s =2 x r,s + y r,s 1/p ≤ 1/q−1/p+1/t p p 2 x r,s + y r,s , where t =min{s, s,r,r} =min{p, q,r,r,s,s} , since s ≥ q (s ≤ q)andp ≥ s. If we instead have s ≥ p,thenby(p, s)-Clarkson and Minkowski inequalities applied to (5.22) we get 1/q 1/s q − q ≤ 1/q−1/s+1/s−1/p+1/t p p s/p x + y r,s + x y r,s 2 ( x r + y r) dμ2 Ω2 p/s p/s 1/p ≤ 1/q−1/p+1/t s s 2 x r dμ2 + y r dμ2 Ω2 Ω2 1/p 1/q−1/p+1/t p p =2 x r,s + y r,s , where t =min{p, s,r,r} =min{p, q,r,r,s,s} , since s ≥ q (s ≤ q)ands ≥ p. Summarizing we can write 1/q 1/p q − q ≤ 1/q−1/p+1/t p p x + y r,s + x y r,s 2 x r,s + y r,s , where t =min{p, q,r,r,s,s} . In order to check that the equality may occur in (5.20) let us take disjoint sets A1,A2 in Ω1 and disjoint sets B1,B2 in Ω2 with corresponding positive ﬁnite measures and deﬁne the functions −1/s −1/r α(u, v)=χA1 (u)χB1 (v)μ1 (A1)μ2 (B1), −1/s −1/r β(u, v)=χA1 (u)χB2 (v)μ1 (A1)μ2 (B2), −1/s −1/r γ(u, v)=χA2 (u)χB1 (v)μ1 (A2)μ2 (B1). 5.3. SOME APPLICATIONS OF THE CLARKSON INEQUALITY 49

Then the required equality occurs on the following pairs of functions 1.x(u, v)=α(u, v),y(u, v)=0; 5.x(u, v)=α(u, v)+γ(u, v), 2.x(u, v)=y(u, v)=α(u, v); y(u, v)=α(u, v) − γ(u, v); 3.x(u, v)=α(u, v),y(u, v)=β(u, v); 6.x(u, v)=α(u, v)+β(u, v), 4.x(u, v)=α(u, v),y(u, v)=γ(u, v); y(u, v)=α(u, v) − β(u, v).

We note, the functions deﬁned with the help of the sets A1,A2,B1 and B2 are real-valued, which means that the best constants in (5.20) are the same in both complex and real case. Observe that by proving inequality (5.20) in Ls[Lr] space we referred each time to the (p, q)-Clarkson inequality in Lr(μ) space. Thus, by induction, the inequality similar to (5.20) is also true in the settings of the mixed norm spaces Lrn [... [Lr1 ] ...] equipped with the quasi-norms (norms when r1, ..., rn ≥ 1) x = ... x ... , where r1, ..., rn > 0. r1,...,rn r1 2 r rn Using inequality (5.20), it is easy to give the following estimation of the modulus of convexity of Ls[Lr] spaces similar to (5.17): if x = y =1and r,s r,s − ≥ ∈ ≤ − p 1/p x y r,s ε, ε [0, 2], then x + y r,s 2 1 (ε/2) and p 1/p p ε 1 ε δ s r (ε) ≥ 1 − 1 − ≥ , where p =max{r, r ,s,s} . L [L ] 2 p 2

This estimation is sharp when 2 ≤ r, s < ∞. The sharp estimate δLs[Lr](ε) ≥ p kr,sε ,wherep =max{r, s, 2} was found by Maleev-Troyanski [37].

Theorem 15. If 0

Theorem 16. Let 0

Thus, Cp,q(X) ≥ Cp,q(R)whenX is a real space (then we take a, b ∈ R)and Cp,q(X) ≥ Cp,q(C)whenX is a complex space (then we take a, b ∈ C). Estimation from above follows by applying twice the triangle inequality for the norm and the property that f(p)=(xp + yp)1/p is decreasing in p (x + yq + x − yq)1/q ≤ 21/q (x + y) ≤ 21/q max{1, 21−1/p} (xp + yp)1/p .

Note that the upper estimate in the above inequality is attained. For 0 1thespaceLr(μ)with1

p q Basic Properties of Constant γ(ln,lm)

In this chapter we consider the relation between complex and real norms of p q bounded linear operators from ln into lm space. The interest to this relation was mainly caused by two reasons. First, we wanted to answer the question if this relation was bounded in the case when 0 sequences x = {xk = f(k)} and integration with summation provided that all integrable functions are everywhere ﬁnite. We supply a space lp,where0

x∞ =sup|xk| =sup|f(k)| = f∞. k∈N k∈N Of course, instead of N we can take the subset 1,n of N and define a finite- p dimensional space ln in terms of a function space. This definition of lp spaces clarifies that all estimates of the constant γ(Lp,Lq) obtained in the previous chapters remain valid with Lp changed to lp spaces. How- ever, we want to improve the general estimates for this particular case. We begin with two important auxiliary lemmas. The techniques we will use in this chapter are not far beyond the scope of the classical Hölder, Minkowski, Jensen inequalities and standard differential calculus.

Lemma 10. Let ak,bk ∈ C for all k ∈ N and 1 ≤ p<∞,then ∞ 1/p ∞ p (6.1) |ak| ≤ |ak|. k=1 k=1

51 p q 52 6. BASIC PROPERTIES OF CONSTANT γ(ln,lm)

{ }∞ The equality holds for p =1if and only if the sequence ak k=1 has at most one element distinct from zero. Proof. ∞ | | ∈ N If k=1 ak = 0 or, alternatively, ak =0forallk , then we clearly ∞ | | ∈ N have the equality in (6.1). Assume that k=1 ak > 0. If we take for any k ∞ p 1/p ck = ak/( |ai| ) , i=1 | | | | ∞ | |p 1/p ≤ ∞ | |p | |≥| |p then ck = ak /( i=1 ai ) 1and k=1 ck = 1. Obviously, ck ck | |≤ ≥ ∞ | |≥ ∞ | |p whenever ck 1and p 1. Thus, k=1 ck k=1 ck =1,thatisequivalent ∞ | | p ≥ ∞ | |p to ( k=1 ak ) k=1 ak . ∞ | | If the equality in (6.1) holds and k=1 ai > 0, then by the preceding ar- | | | |p ∈ N gument we should have ck = ck for all k , that together with the re- ∞ | |p ∈ N | | lation k=1 ck = 1 implies that it should exist k such as 1 = ck = | | ∞ | |p 1/p ak /( i=1 ai ) and all other elements ck should be equal to zero. Now it is { }∞ easy to understand that in order to satisfy this condition the sequence ak k=1 may have at most one element distinct from zero. Lemma . ≤ ≤∞ ∈ p 11 If 0

Note, for p

Recall, for bounded linear operators T between Lebesgue spaces Lp(μ)and Lq(ν), where μ and υ are arbitrary σ-ﬁnite measures we have the following state- p q ments. The equality TC = T holds for all T ∈L(L (μ),L (υ)) if 0

Proposition . ≤∞ p → q 9 Let 0

p q The results on constant γ(ln,lm) which will be obtained in this chapter can be brieﬂy summarized in the following theorem. Theorem 17. Let n, m ∈ N, n, m ≥ 2. ≤∞ p q ≤ (i) If 0 1. p q ≤ ≥ (iv) γ(l2,l2)=1if and only if either 0

0 ∈L p q for which we can claim that the norm of any operator T (l2,l2)coincideswith the norm of its complexiﬁcation TC. The ﬁrst proposition is a simple generalization of the statement of Verbicki˘ı-Sereda [59]. Proposition . ∈ N ≥ ≥ 10 Let n and n 2. If q 2,then TC p,q = T p,q ∈L p q ≤ ≤ ≥ for any T (l2,ln).If1 p 2 and q 1, then TC p,q = T p,q for any ∈L p q ∈L p q T (ln,l2).If 0

Proof. Assume the operator T is deﬁned by the rule T (x, y)=(a1,1x + a1,2y, ..., an,1x + an,2y) and let the norm of TC be attained at the vector v =(z,w) with z,w ∈ C.Ifz =0,thenv = w(0, 1) (w =0)and

TCvq TC(w(0, 1))q |w|TC(0, 1)q TC(0, 1)q TC =sup = = = , p,q | | vp=0 v p w(0, 1) p w (0, 1) p (0, 1) p that shows the norm of TC is also attained at the real vector (0, 1) and thus TCp,q = T p,q. The same observation says that w =0 . Now, for z =0 take v = z(1,w/z) and note TC(z(1,w/z)) q = TC(1,w/z) q . Assume that w/z = z(1,w/z)p (1,w/z)p ρ(cos ϕ + i sin ϕ), then TC(1,ρ(cos ϕ + i sin ϕ))q is equal to

(a1,1 + a1,2ρ cos ϕ + ia1,2ρ sin ϕ, ..., an,1 + an,2ρ cos ϕ + ian,2ρ sin ϕ)q q 1 q =(|a2 +2a a ρ cos ϕ + a2 ρ2|q/2 + ... + |a2 +2a a ρ cos ϕ + a2 ρ2| 2 ) . 1,1 1,1 1,2 1,2 n,1 n,1 n,2 n,2

Consider the function φρ(ϕ) deﬁned for every ﬁxed ρ on the interval [0,π)by (6.4) q q φ (ϕ)=(a2 +2a a ρ cos ϕ + a2 ρ2) 2 + ... +(a2 +2a a ρ cos ϕ + a2 ρ2) 2 . ρ 1,1 1,1 1,2 1,2 n,1 n,1 n,2 n,2 1/q Note, that TC(1,ρ(cos ϕ + i sin ϕ)) q = φρ(ϕ) and thus, the norm of TC is attained at the same vector at which the function φρ(ϕ) has its maximum. Since the function φρ(ϕ) is even and periodic with period 2π, it is enough to consider its values on the interval [0,π). The derivative of φρ(ϕ)is q − − 2 2 2 2 1 φρ(ϕ)= qρsin ϕ[a1,1a1,2(a1,1 +2a1,1a1,2ρ cos ϕ + a1,2ρ ) + ... q 2 2 2 2 −1 +an,1an,2(an,1 +2an,1an,2ρ cos ϕ + an,2ρ ) ].

We note φρ(ϕ) can have its maximum on [0,π)ateitherϕ1 =0orϕ2 = π ∈ 2 or ψ(ϕ3)=0, where ϕ3 (0,π). Consider the functions ψk(ϕ)=ak,1ak,2(ak,1 + 2 2 q/2−1 ≥ 2ak,1ak,2 cos ϕ+ak,2ρ ) for k =1 , ..., n. For q 2 these functions are decreasing n on [0,π) and thus the sum ψ(ϕ)= k=1 ψk(ϕ) is also a decreasing function. If there ∈ exists ϕ3 (0,π) such that ψ(ϕ3) = 0 then it is easy to check that φρ(ϕ) changes its sign from minus to plus and hence φρ(ϕ) decreases first and then increases on [0,π) which also means that ϕ3 should be the point of minimum of φρ(ϕ). Therefore φρ(ϕ) may attain its maximum only at the end points of the interval [0,π)or,in other words, at the real vector and therefore TCp,q = T p,q. The second statement of this proposition can be obtained by duality from the first one which we have just proved. ∈L p q As an example of the operator F (l2,l2)forwhich FC p,q = F p,q we can take F (x, y)=(x + y,x − y). Real and complex norms of this operator differ for the case when 0

Problem . p → q ≥ 5 Take any operator T : l3 l2 and p>q 2. Is it true that TC = T ? Recall that Riesz example 3 shows that for all 0 T . In the proof of Proposition 10 we used the duality argument. In the next lemma we obtain this statement directly and what is the more important we extend the result on the equality of complex and operator norms to the quasi-Banach case. Proposition . ∈L p q ≤ 11 Let T (l2,l2).If0 0andd<0. If the operator T attains its norm at the complex vector (z1,z2). Then, T also attains its norm at (1,z), where z = ρ(cos θ + i sin θ) ∈ C and we will show that either θ =0or θ = π or ρ =0orρ = ∞ that will imply the equality T p,q = TC p,q. For 0

2 a2 +2abρ cos θ + b2ρ2 cd q−2 (6.8) = − = α. c2 +2cdρ cos θ + d2ρ2 ab If condition (6.7) is satisﬁed, then we get what we wanted. Thus, only relation (6.8) can contribute θ which diﬀers from 0 and π. In this case if we had c2 +2cdρ cos θ + d2ρ2 =0, then it should follow for abcd < 0thata2 +2abρ cos θ + b2ρ2 =0and therefore TC(1,z) =0, that is clearly not possible. Now, Q1/q−1 F (ρ, θ)= [(a2 +2abρ cos θ + b2ρ2)q/2−1(ab cos θ + b2ρ)P ρ P 1/p+1 +(c2 +2cdρ cos θ + d2ρ2)q/2−1(cd cos θ + d2ρ)P − Qρp−1] Q1/q−1 = ((a2 +2abρ cos θ + b2ρ2)q/2−1 × P 1/p+1 (ab cos θ + abρp cos θ + b2ρ + b2ρp+1 − a2ρp−1 − 2abρp cos θ − b2ρp+1) +(c2 +2cdρ cos θ + d2ρ2)q/2−1 × (cd cos θ + cdρp cos θ + d2ρ + d2ρp+1 − c2ρp−1 − 2cdρp cos θ − d2ρp+1)) p q 56 6. BASIC PROPERTIES OF CONSTANT γ(ln,lm)

Q1/q−1 = ((a2 +2abρ cos θ + b2ρ2)q/2−1(ab(1 − ρp)cosθ + b2ρ − a2ρp−1) P 1/p+1 +(c2 +2cdρ cos θ + d2ρ2)q/2−1(cd(1 − ρp)cosθ + d2ρ − c2ρp−1)). Hence, Fρ(ρ, θ)=0if (a2 +2abρ cos θ + b2ρ2)q/2−1(ab(1 − ρp)cosθ + b2ρ − a2ρp−1) +(c2 +2cdρ cos θ + d2ρ2)q/2−1(cd(1 − ρp)cosθ + d2ρ − c2ρp−1)=0. Therefore, the extremum points are defined by (a2 +2abρ cos θ + b2ρ2)q/2−1 cd(1 − ρp)cosθ + d2ρ − c2ρp−1 (6.9) = − . (c2 +2cdρ cos θ + d2ρ2)q/2−1 ab(1 − ρp)cosθ + b2ρ − a2ρp−1 The combination of conditions (6.7) and (6.9) implies that the maximum of F (ρ, θ) can be attained at θ =0orθ = π and ρ defined by the relation (6.9), i.e. on the boundary of [0, ∞] × [0,π] and we are done, otherwise due relation (6.8) we should have the point of extremum (ρ, θ) defined by cd cd(1 − ρp)cosθ + d2ρ − c2ρp−1 (6.10) = . ab ab(1 − ρp)cosθ + b2ρ − a2ρp−1 1 − bd p−2 ≤ For p = 2 this equality is satisfied for ρ1 = ac = 0 and for 1

1 1 − bd(adα−bc) p−2 − bd p−2 Thus, ρ1 =( ac(ad−bcα) ) or ρ2 =0. We note that ρ1 coincides with ac 1 − − bd(adα−bc) p−2 for either α =1orad = bc. Moreover, for p<2andρ =( 2 ac(ad−bcα) ) < 1 − bd(adα−bc) p−2 ρ1 we have that f (ρ) < 0andifρ =( 2ac(ad−bcα) ) >ρ1,thenf (ρ) > 0. Therefore, if ρ and θ are connected by relation (6.8) then the function f(ρ)ﬁrst ≤ ≤ decreases and then increases that implies F ρ1,θρ1 F (0,θ)orF ρ1,θρ1 p q 6. BASIC PROPERTIES OF CONSTANT γ(ln,lm)57

F (∞,θ) or, in other words, there exists some point (ρ, θ) on the boundary for which ≥ F (ρ, θ) F ρ1,θρ1 and hence F ρ1,θρ1 is not the point of global maximum of F (ρ, θ). In the case when p =2andρ = 0 to satisfy (6.10) we should have ac = −bd and thus 2 2 2 2 2 2 −1/2 2 (c +d ρ )α−(a +b ρ ) 2 2 a + ab ab−cd + b ρ f (ρ)= ρbd(α − 1) [ad + bc] . (1 + ρp)1/p+1(ab − cd)1/2 Hence, if α =1and ad + bc =0then f (ρ) = 0 if and only if ρ =0. Otherwise, if α =1orad + bc =0,thenf (ρ) = 0 for every ρ or f(ρ) is constant in ρ and thus F (ρ, θ)=F (0,θ). Hence, the global maximum of F is attained on the boundary. Therefore the norm of the operator T is attained at the real vector and T p,q = TC p,q.

Remark 22. If 0

Proposition 11 show that F (ρ1,θρ1 ) deﬁnes the norm of the operator T, where

1 2 bd p−2 (c2 + d2ρ2)α − (a2 + b2ρ2) cd q−2 ρ = − and cos θ = 1 1 with α = − . 1 ρ1 − ac 2ρ1(ab cd) ab Summarizing the results of Propositions 10 and 11 we have

Theorem 18. If either 0

The following proposition gives the estimate of the relation between complex ∈L p q and real norms of any operator T (l2 ,l2) in the case not covered by Theorem 18. Proposition . ∈L p q ≥ ≤ ≤ 12 Let T (l2,ln) and n 2.If0

1/q−1/2 TCp,q ≤ min{2, 2 }T p,q.

1/q−1/2 1/p−1/2 Moreover, if 1 ≤ q ≤ 2 ≤ p,thenTCp,q ≤ min{2 , 2 }T p,q. p q 58 6. BASIC PROPERTIES OF CONSTANT γ(ln,lm)

Proof. Let operator T be given by n × 2matrixA =(ai,j ). Then for z1 = iθ1 iθ2 r1e and z2 = r2e we get n q | |q TC(z1,z2) q = ak,1z1 + ak,2z2 k=1 n iθ1 iθ2 q = |ak,1r1e + ak,2r2e | k=1 n iθ1 i(θ2−θ1) q = |e ||ak,1r1 + ak,2r2e | k=1 n 2 2 − 2 2 q/2 = (ak,1r1 +2ak,1ak,2r1r2 cos(θ2 θ1)+ak,2r2) k=1 2 2 In the trigonometric identity cos(θ)=cos(θ/2) − sin (θ/2) with θ = θ2 − θ1 we denote cos2(θ/2) by α and sin2(θ/2) by β. Clearly, α + β =1andα, β ≥ 0. Then, n 2 2 − 2 2 q/2 (ak,1r1 +2ak,1ak,2r1r2 cos(θ2 θ1)+ak,2r2) k=1 n 2 2 2 2 = α(ak,1r1 +2ak,1ak,2r1r2 + ak,2r2) k=1 q/2 2 2 − 2 2 + β(ak,1r1 2ak,1ak,2r1r2 + ak,2r2)

n 2 2 q/2 = [α(ak,1r1 + ak,2r2) + β(ak,1r1 − ak,2r2) ] k=1 (1) n n q/2 q q/2 q ≤ α |ak,1r1 + ak,2r2| + β |ak,1r1 − ak,2r2| k=1 k=1 (2) ≤ q/2 q/2 q p p q/p (α + β ) T p,q(r1 + r2 ) q/2 q/2 q | |p | |p q/p =(α + β ) T p,q( z1 + z2 ) (3) ≤ 1−q/2 q | |p | |p q/p 2 T p,q( z1 + z2 ) . We should mention here that inequality (1) holds for q ≤ 2 by Lemma 11. Bound- edness of T implies inequality (2) and from H¨older inequality it follows inequal- 1/q−1/2 ity (3). Hence, TCp,q/T p,q ≤ 2 . By Theorem 17 (i) we also have 1/q−1/2 TCp,q/T p,q ≤ 2. Therefore, TCp,q/T p,q ≤ min{2, 2 }. The second statement can be obtained from just proved by duality. Corollary 6. Let n ∈ N, n ≥ 2. ≤ ≤ ∞ q 1/q−1/2 (i) If 1 q 2,thenγ(l2 ,ln)=2 . ≥ p 1 1/p−1/2 (ii) If p 2,thenγ(ln,l2)=2 .

Proof. (i) Consider the operator F given by n × 2matrix(ai,j )witha11 = a12 = a21 = −a22 = 1 and all other elements aij are equal to zero. Applying ∞ this operator to any vector in l2 results in n-dimensional vector with at most two p q 6. BASIC PROPERTIES OF CONSTANT γ(ln,lm)59 nonzero components. It allows us to consider the operator F as an operator between two-dimensional lp spaces and hence using the results of Theorems 10 and 11 from 1/q−1/2 Chapter 5 we conclude that FC∞,q/F ∞,q =2 . That means the estimate 1/q−1/2 TC∞,q/T ∞,q ≤ 2 of Proposition 12 cannot be improved in this case and ∞ q 1/q−1/2 consequently we have γ(l2 ,ln)=2 . The second statement follows from (i) by duality. Problem . ∈L p q − 6 The operator T (l2 ,l2) given by T (x, y)=(x + y,x y) is ≤∞ p q extremal in the sense that for 0

CHAPTER SEVEN

Riesz-Thorin Interpolation Theorem

In this chapter we collect some well known results concerning the Riesz-Thorin interpolation theorem in the complex case and show how the knowledge of the relation between complex and real operator norms helps to get the real version of this theorem from the complex one. It seems that this method of getting the real Riesz-Thorin interpolation theorem is the simplest one if we are careful of the so called Riesz constant. We also present here some new, as we hope, information concerning the real version of this theorem for the case when at least one of the spaces is quasi-Banach. Thus, we treat both real and complex versions of the Riesz-Thorin interpolation theorem in the whole first quadrant. Moreover, we draw some conclusions concerning the real Riesz-Thorin interpolation theorem for the operators between certain finite dimensional lp spaces and between mixed norm Lebesgue spaces. There are different ways to state the Riesz-Thorin interpolation theorem and here we follow the classical one (cf. Bergh-Löfström [4], Bennett-Sharpley [3], Krein-Petunin-Semenov [32], Zygmund [65]). Definition 8. Afunctionf defined on the measure space (Ω,μ) is said to be simple if it takes only finite number of values. If, additionally, μ(Ω) = ∞, then we also assume that function f vanishes outside a subset of Ω of finite measure. We will denote the set of simple functions by S.

Definition 9. Let (Ω1,μ) and (Ω2,ν) be σ-ﬁnite measure spaces and let operator T be deﬁned on all μ-simple functions on Ω1 and resulting in the ν-measurable functions on Ω2.Suppose0

(7.1) TfLq(ν) ≤ MfLp(μ), for all μ-simple functions f on Ω1,thenT is said to be of strong type (p, q).The least constant M for which (7.1) holds is called the strong type (p, q) norm of T (or simply the norm of T if no confusion can arise).

If 0

61 62 7. RIESZ-THORIN INTERPOLATION THEOREM

Theorem 19 (Riesz-Thorin interpolation theorem). Let (Ω1,μ) and (Ω2,ν) be two measure spaces and T be a linear operator deﬁned for all simple complex-valued functions f on (Ω ,μ).Let0

If T is simultaneously of strong types (p0,q0) and (p1,q1) with strong type constants M0 and M1 respectively, then T is also of strong type (p, q) and its strong type norm Mθ satisfies ≤ 1−θ θ (7.3) Mθ M0 M1 . For the sake of completeness we put here the proof of this theorem given by Calderón and Zygmund in [10]. Proof. Let k>0besosmallthat k k (7.4) < 1, < 1, q0 q1 and let p and q be given by (7.2). We observe that relations (7.2) and (7.4) imply k/q < 1. Hence, (7.5) k | |k | |k Tf q = Tf q/k =sup Tf gdν : g is simple and g 1/(1−k/q) =1 . g Ω2 ≥ | | iu We may assume that f p =1andg 0. Let fix f and g,writef = f e and consider the integral I = |Tf|kgdν. Ω2 Consider also the functions p(z)andq(z) defined by 1 (1 − θ) θ 1 (1 − θ) θ = + and = + , p(z) p0 p1 q(z) q0 q1 which for z =0andz = 1 reduce to p0,q0 and p1,q1 respectively. Moreover, p(θ)=p and q(θ)=q. Suppose p<∞. We introduce the simple functions 1−k/q(z) p/p(z) iu 1−k/q Fz = |f| e ,Gz = g depending on the parameter z and the integral k (7.6) Φ(z)= |TFz| |Gz|dν, Ω2 ≥ which reduces to I for z = θ (since g 0). If c1,c2, ... and c1,c2, ... be the various values taken by the functions f and g respectively, and let χ1,χ2, ... and χ1,χ2, ... be the characteristic functions of the sets where they are taken, then writing cj = iuj |cj|e , we have for 0 ≤ x ≤ 1, | | iuj f = cj e χj,g= clχl, 1−k/q(z) iuj | |p/p(z) 1−k/q Fz = e cj χj,Gz = (cl) χl and iuj | |p/p(z) TFz = e cj Tχj. 7. RIESZ-THORIN INTERPOLATION THEOREM 63

z It is easy to see that Gz and TFz are linear combinations of functions λ with k λ>0 and with coeﬃcient functions deﬁned on Ω2. Thus |TFz| |Gz| for every point in Ω2 is a continuous and subharmonic function in z = x + iy for 0 ≤ x ≤ 1andso also Φ(z). The function Φ(z) is also bounded there. We note, k k | |k iuj | |p/p(z) ≤ · TFz = e cj Tχj const Tχj ,

k 1−k/q(z) | | 1−k/q ≤ · Gz = (cl) χl const χl, | |≤ · k · k (7.7) Φ(z) const TCχj χldν = const TCχj dν, Ω2 Ω2,l where Ω2,l is a subset of Ω2 where χl =0. Thus Ω2,l is of ﬁnite measure. Taking k so small that k

Then, by the H¨older inequality and (7.8) we have for all positive q0 and q1 q(1−θ) qθ q0 q | |q | |q(1−θ) | |qθ q1 Tf q = Tf dν = Tf Tf dν Ω2 Ω2 q(1−θ) qθ q(1−θ) qθ ≤|Tf| q0 |Tf| q1 = Tf Tf q(1−θ) qθ q0 q1 q(1−θ) qθ | |q = M0 M1 esssup f and the proof is complete.

Remark 23. If 0

This can be one of the reasons why we consider the Riesz-Thorin interpolation theorem in the whole ﬁrst quadrant. Another reason is, the H¨older inequality shows that every operator of strong type (p, q) is also of strong type (p, q1) for 0 0: T p,q γ T p0,q0 T p1,q1

for all T of strong types (pi,qi),i=0, 1}, where p and q are given by (7.2), 0 <θ<1and0

Theorem 20 (Riesz-Thorin interpolation theorem in the real case). Let (Ω1,μ) and (Ω2,ν) be two measure spaces and T be a linear operator deﬁned for all simple functions f on (Ω ,μ).Let0

Figure 3. Real Riesz constant. ≤∞ ≤ (i) If 0

d d dp0 q0 1−θ dp1 q1 θ (iii) If 1 ≤ pi,qi ≤∞, i =0, 1,thenρ(θ, pi,qi) ≤ min{ , } min{ , } . dq0 dp0 dq1 dp1 (iv) If 0

Proof. All these estimates of the constant ρ(θ, pi,qi) can be obtained from inequality (7.9), Theorems 8, 9 and Proposition 7.

Remark 24. If pi >qi for some i ∈ 0, 1 (one point in the ”upper triangle”), ≤ 1−θ θ then the inequality T p,q T p0,q0 T p1,q1 does not hold in general. Example . 11 6 Take operator T (x, y) deﬁned by matrix − . As it was √ √1 1 calculated early T ∞,1 =2, T 2,2 = 2 and T 4,4/3 = 3 (see Theorem 10 and the consequence of Theorem 11 from Chapter 5). In the settings of Riesz- Thorin interpolation theorem 3/4=(1− 1/2) · 1/2+1/2 · 1 and θ =1/2. But √ √ 1/2 1/2 3/4 1/2 1/2 T 4,4/3 = 3 > 2 2 =2 = T 2,2 T ∞,1 . Moreover, Thorin [58] ≥ showed that T p,p is logarithmically concave as a function of 1/p for p 2.

Remark 25. It is not possible to replace the condition pi ≤ qi for i =0, 1 by p ≤ q in the real Riesz-Thorin interpolation theorem. 1 −1 Example 7 (Vogt [61]). Take operator T given by matrix with 11.1 p0 = ∞,q0 =1,p1 =3/2,q1 =3and θ =3/4. Then, the relation T p,q ≤ 1−θ θ T p0,q0 T p1,q1 does not hold with this modiﬁed condition. Remark 26. In most applications of the Riesz-Thorin theorem it is not necessary to know the exact value of the Riesz constant. However, sometimes this knowledge is very important, as it is in the interpolation proof of the classical Clarkson inequality from which the uniform convexity of Lp spaces for 1 ≤ p<∞ follows. Using Theorems 8, 9, 17 and Proposition 9 we can derive in similar way the real Riesz-Thorin interpolation theorem for bounded linear operators between ﬁnite- p − q dimensional ln lm spaces from the complex Riesz-Thorin interpolation theorem with better Riesz constant for some values of p, q and n, m than those in (7.3). We can also obtain real version of the Riesz-Thorin interpolation theorem for the operator between mixed norm Lebesgue spaces from its complex analogue proved by Benedek-Panzone [2] and Theorems 2 and 7.

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