Isomorphic classification of weak lp spaces

ISOMORPHIC CLASSIFICATION OF WEAK LP SPACES RUDY SABARUDIN (M.Sc., National University of Singapore) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2006 Acknowledgements I would like to thank my thesis supervisor, Denny Leung, whose ideas inspired this thesis. I am really grateful for his friendly guidance and great patience during my five-year postgraduate study at NUS. Uncountable thanks to my beloved wife, Huiping, for her love and sacrifice. Thanks to friends and band-mates from ARPC and Hengki Tasman from ITB, who remember me in their prayers, to NUS ex-classmates for the moral support, especially for Toh Pee Choon, who attended my oral defense and helped me with technical matters, and to colleagues at Temasek Polytechnic for their support in the preparation of my oral QE. And most of all, to my Master for His faithful love and grace. Soli Deo Gloria. ”Thank You LORD for saving my soul, Thank You LORD for making me whole, Thank You LORD for giving to me Thy great salvation so rich and free.” Rudy Sabarudin December 2006 ii Contents Acknowledgements ii Summary v Introduction 1.1 Definitions, Notations, and Basic Results . . . . . . . . . . . . . . . . . . 1.2 The Objectives and Outline of the Thesis . . . . . . . . . . . . . . . . . . Subspace Structure of Lp,∞ {−1, 1}I 2.1 Embeddings from Lp,∞ Ji into Lp,∞ {−1, 1}I . . . . . . . . . . . . 10 into Lp,∞ ({−1, 1}I ) . . . . . . . . . . . . . . . . 30 i∈I 2.2 Embedding from p,∞ (Γ) Embeddings Between Lp,∞ Spaces 39 3.1 Representation of Purely Non-atomic Lp,∞ Spaces . . . . . . . . . . . . . 40 3.2 Isomorphic-invariant Parameters . . . . . . . . . . . . . . . . . . . . . . . 45 3.3 Characterization of An Embedding Between Weak Lp Spaces, p ≥ . . . 52 Isomorphic Classification of Purely Non-Atomic Weak Lp Spaces 57 4.1 The Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2 The Uniqueness of Isomorphic Classes . . . . . . . . . . . . . . . . . . . . 69 iii Contents Bibliography iv 76 Summary The objective of this thesis is to present the isomorphic classification of Lp,∞ (Ω, Σ, µ) spaces, < p < ∞, for purely non-atomic measure spaces (Ω, Σ, µ). Chapter gives a brief introduction on the terminologies and notations related to weak Lp spaces . This will be given in Section 1.1 together with some useful results by Carothers and Dilworth [1], [3], and Leung [9]. In Section 1.2, we review the development on the study of isomorphic problems of weak Lp spaces and state the objectives and outline of the thesis. In Chapter 2, Leung’s result in [7] will be extended for some purely non-atomic measure spaces. In particular, we will observe the subspace structure of Lp,∞ ({−1, 1}I ) for arbitrary index set I. Embeddings from Lp,∞ ( studied in Section 2.1, while embeddings from I [0, 1]) p,∞ (Γ) into Lp,∞ ({−1, 1}I ) will be into Lp,∞ ({−1, 1}I ), |Γ| > ℵ0 , in Section 2.2. Embeddings between certain weak Lp spaces will be studied in Chapter 3. First, we present a representation of purely non-atomic weak Lp spaces in Section 3.1. Then followed by introducing three isomorphic-invariant parameters on purely non-atomic weak Lp spaces in Section 3.2. And finally, in Section 3.3 we give the characterization of complemented embeddings between certain purely non-atomic weak Lp spaces. The classification theorem will be presented in Chapter 4. In Section 4.1, it will be v Summary shown that for any non-atomic measure spaces (Ω, Σ, µ), the weak Lp spaces Lp,∞ (Ω, Σ, µ) can be classified into two mutually exclusive groups (for p ≥ 2) with nine isomorphic classes in each group. Section 4.2 closed the chapter with partial results and open problems on the uniqueness of the classification. vi Chapter Introduction In this chapter we introduce some common notations and terminologies in Banach space theory, mostly follow those in [10] and [11]. Notations related to weak Lp spaces and measure spaces will also be given. In Section 1.2, we shall see the development of the study of isomorphism between weak Lp spaces. We close this chapter with the objectives and outline of the thesis. 1.1 Definitions, Notations, and Basic Results A weak Lp space is a Lorentz function space Lp,q with q = ∞. For this reason, the weak Lp spaces will be denoted by Lp,∞ . Here is the definition. Definition 1.1. Let (Ω, Σ, µ) be a measure space. For < p < ∞, the weak Lp space Lp,∞ (Ω, Σ, µ) is the space of all (equivalence classes of) Σ-measurable functions f such that f = sup c(µ{|f | > c})1/p < ∞. (1.1.1) c>0 In many literature, the quasi-norm f p,∞ · defined in (1.1.1) is written as = sup t1/p f (t), (1.1.2) t>0 where f is the decreasing rearangement of |f | defined by f (t) = inf{s > : µ{|f | > s} ≤ t}, ≤ t < µ(Ω). 1.1 Definitions, Notations, and Basic Results Using substitution t = µ{|f | > c} for any c > 0, it is clear that equivalent. Note that · p,∞ · and · p,∞ are defined in (1.1.2) is not a norm on Lp,∞ (Ω, Σ, µ) because it does not satisfy triangle inequality. However, it is well known that · p,∞ is equivalent to a norm under which Lp,∞ (Ω, Σ, µ) is a Banach space. We refer to [13, Chapter 12] for the details. This norm is given by |||f ||| = sup A∈Σ A |f | dµ , 1−1/p (1.1.3) µ(A) where the supremum is taken over all measurable set A with < µ(A) < ∞. Since we are only concerned with isomorphic questions, the quasi-norm · in (1.1.1) shall be used in most computations involving norm on Lp,∞ spaces. For an arbitrary set Ω and any G ⊆ Ω, we define the indicator function of G, 1G , by   1 if ω ∈ G; 1G (ω) =  0 otherwise. Below are notations for some special Lp,∞ spaces and subspaces. = Lp,∞ (Ω, P(Ω), µ), where µ is the counting measure on the power p,∞ (Ω) set P(Ω). p,∞ = p,∞ (N). M p,∞ (Ω, Σ, µ) = the closed subspace of Lp,∞ (Ω, Σ, µ) generated by the indicator Lp functions 1σ , σ ∈ Σ with µ(σ) < ∞. space Lp,∞ (Ω, Σ, µ) is a Banach lattice with the natural ordering ≤ Every weak defined by f ≤ g if f (ω) ≤ g(ω) for µ-a.e. ω ∈ Ω. The binary operations ∨ and ∧ are defined by f ∨ g = sup{f, g} and f ∧ g = inf{f, g}. Two elements f and g in a Banach lattice is disjoint if |f | ∧ |g| = 0. Definition 1.2. Let X be a Banach space. Y is a complemented subspace of X if there is a projection P on X such that P (X) = Y . For Banach spaces X and Y , we shall use the following notations. X∼Y = X is isomorphic to Y . X∼ =Y = X is isometrically isomorphic to Y . X →Y = X is isomorphic to a subspace of Y . c X →Y = X is isomorphic to a complemented subspace of Y . 1.1 Definitions, Notations, and Basic Results The following theorem is a well known variant of Pelczy´ nski’s ”decomposition method”. For a proof, see [7]. Theorem 1.3. Let X and Y be Banach spaces. Suppose X ∼ (X ⊕X) and Y ∼ (Y ⊕Y ). c c If X → Y and Y → X, then X ∼ Y . Lp,∞ (Ω, Σ, µ) is the dual space of Lq,1 (Ω, Σ, µ), where p + q = 1. The norm on the Banach space Lq,1 (Ω, Σ, µ) is given by ∞ g q,1 t1/q−1 g (t)dt. = Now we recall some terminologies regarding measure spaces. A measurable subset σ of a measure space (Ω, Σ, µ) is an atom if µ(σ) > 0, and either µ(σ ) = or µ(σ \ σ ) = for each measurable subset σ ⊂ σ. A purely non-atomic measure space is one which contains no atoms. A collection S of measurable sets generates a measure space (Ω, Σ, µ) if Σ is the smallest σ-algebra containing S as well as the µ-null sets. A purely atomic measure space is one which is generated by the collection of all of its atoms. A countably generated measure space is one which is generated by a sequence of measurable sets. A measure space (Ω, Σ, µ) is said to be σ-finite if there is a sequence (Ωn )n∈N of measurable sets with finite measure such that Ω = Let (Ω1 , Σ1 , µ1 ) and Ωn . n∈N (Ω2 , Σ2 , µ2 ) be measure spaces. Denote by Nµ1 and Nµ2 the µ1 − and µ2 −null sets respectively. Then µ1 (respectively µ2 ) induces a function µ ˆ1 (respectively µ ˆ2 ) on the σ-complete Boolean algebra Σ1 /Nµ1 (respectively Σ2 /Nµ2 ). Definition 1.4. Two measure spaces (Ω1 , Σ1 , µ1 ) and (Ω2 , Σ2 , µ2 ) are isomorphic if there is a Boolean algebra isomorphism Φ : Σ1 /Nµ1 → Σ2 /Nµ2 such that µ ˆ1 = µ ˆ2 ◦ Φ. The following theorem, which can be found in [12], follows from the observation that the collection of functions of the from an 1An , where the sum is taken pointwise, n∈N (an ) ⊂ R and (An )n∈N is a pairwise disjoint sequence in Σ, is dense in Lp,∞ (Ω, Σ, µ). Theorem 1.5. If (Ω1 , Σ1 , µ1 ) and (Ω2 , Σ2 , µ2 ) are isomorphic measure spaces, then Lp,∞ (Ω1 , Σ1 , µ1 ) ∼ = Lp,∞ (Ω2 , Σ2 , µ2 ). 1.1 Definitions, Notations, and Basic Results Let (Ω, Σ, µ) be a measure space and let < a ∈ R. Define the measure space a · (Ω, Σ, µ) as (Ω, Σ, a · µ), where (a · µ)(σ) = a µ(σ) for all σ ∈ Σ. Let A be an index set and let (Ωα , Σα , µα )α∈A be pairwise disjoint measure spaces. The measurable space (Ω, Σ) = (Ωα , Σα ) is defined to be the set Ω = α∈A Ωα endowed with the smallest α∈A Σα . For any σ ∈ Σ, define σ-algebra Σ generated by α∈A µα (σ ∩ Ωα ) = µ(σ) = µα (σ ∩ Ωα ) . sup finite F ⊆A α∈A α∈F (Ωα , Σα , µα ). In this measure space, σ ∈ Σ The measure space (Ω, Σ, µ) is denoted by α∈A if and only if (σ ∩ Ωα ) ∈ Σα for all α ∈ A and there exists countable B ⊆ A such that for all α ∈ / B, σ ∩ Ωα are all equal to Ωα or all empty. The support of a real-valued function f on Ω is defined by {ω ∈ Ω : f (ω) = 0}. The following proposition states that the support of a function f ∈ Lp,∞ (Ωα , Σα , µα ) α∈A is contained a.e. in a countable union of Ωα . Proposition 1.6. For every f ∈ Lp,∞ (Ωα , Σα , µα ) , f α∈A Ωα = a.e. except for countably many α ∈ A. Proof. Without loss of generality, assume |A| > ℵ0 . Let f ∈ Lp,∞ (Ωα , Σα , µα ) . α∈A Then µ{|f | > c} < ∞ for all c > 0. Suppose there is an uncountable subset A ⊆ A with f Ωα = a.e. for all α ∈ A . If for every δ > 0, α ∈ A : µα f 1Ωα > δ = is countable, then ∞ ℵ0 ≥ α ∈ A : µα f 1Ω α > n=1 n =0 = α ∈ A : µα |f 1Ωα | > = ≥ |A |, a contradiction. Hence, there exists δ > with µα {|f 1Ωα | > δ} = for uncountably many α ∈ A. So, there is > and an infinite subset A ⊆ A so that µα |f 1Ωα | > δ > for all α ∈ A . But this implies µ{|f | > δ} = ≥ sup finite F ⊂A which is impossible. α∈F µα |f 1Ωα | > δ sup finite F ⊂A ≥ µα |f 1Ωα | > δ sup finite F ⊂A α∈F |F | = ∞, 4.1 The Classification 62 On the other hand, Lp,∞ aβn · {−1, 1}Iβn Lp,∞ an · {−1, 1}Iβn n∈K c →Lp,∞ n∈K aα · {−1, 1}Iα   Lp,∞  aα · {−1, 1}Iα  α∈A c →Lp,∞ n∈K βn for all α ∈ A and |Iα | ≤ |Iα | whenever aα = ∞ for all γ ∈ A, where Aγ = {α ∈ A : |Iα | ≥ |Iγ |}. Then α < α . Suppose α∈Aγ 1. If {|Iα | : α ∈ A} has a maximum, say Iα0 , then Lp,∞ aα · {−1, 1}Iα α∈A where |Jn | = Iα0 for all n ∈ N; ∼ Lp,∞ {−1, 1}Jn n∈N , 4.1 The Classification 63 2. If {|Iα | : α ∈ A} has no maximum, then Lp,∞ aα · {−1, 1}Iα ∼ Lp,∞ {−1, 1}Jn α∈A , n∈N where ℵ0 ≤ |J1 | < |J2 | < · · · . Proof. First, we consider the case where {|Iα | : α ∈ A} has a maximum, say Iα0 . For each α ∈ A, let Fα be a finite index set such that aα ≤ |Fα |. Then for each α ∈ A, the {−1, 1}Iα0 measure algebra of aα · {−1, 1}Iα is isomorphic to a measure sub-algebra of Fα and hence, c Lp,∞ aα · {−1, 1}Iα → Lp,∞ {−1, 1}Iα0 . Fα Since A is countable, we have a bijection from Fα onto N. Thus, α∈A Lp,∞ aα · {−1, 1}Iα c → Lp,∞ α∈A {−1, 1}Iα0 α∈A Fα c → Lp,∞ {−1, 1}Jn , (4.1.4) n∈N where |Jn | = Iα0 for all n ∈ N. aα = ∞, for each n ∈ N, there is a finite subset On the other hand, since α∈Aα0 Gn ⊂ Aα0 such that α∈Gn aα ≥ and (Gn )∞ n=1 is pairwise disjoint. Then, for each n ∈ N, the measure algebra of {−1, 1}Jn is isomorphic to a measure sub-algebra of aα · {−1, 1}In and hence, α∈Gn Lp,∞ {−1, 1}Jn c → Lp,∞ aα · {−1, 1}In n∈N α∈Gn n∈N   → Lp,∞  an · {−1, 1}In  c α∈Aα0 c an · {−1, 1}In → Lp,∞ . (4.1.5) α∈A By Theorem 1.3, (4.1.4) and (4.1.5), we have Lp,∞ aα · {−1, 1}In α∈A ∼ Lp,∞ {−1, 1}Jn n∈N . 4.1 The Classification 64 Next, we consider the case where {|Iα | : α ∈ A} has no maximum. Choose a sequence α1 < α2 < α3 < · · · in A such that |Iα1 | < |Iα2 | < |Iα3 | < · · · and sup |Iαn | = sup |Iα |. α∈A n∈N For each n ∈ N, let Jn = Iαn . Let (Sn )∞ n=1 be a partition of N with |Sn | = ℵ0 for all n ∈ N and for each n ∈ N, let the elements of Sn be ordered with the usual ordering of N. Since A is countable, there exists a bijection α ↔ nα , nα ∈ N. And for each α ∈ A, there is a finite subset Fα ⊂ Snα such that |Iα | ≤ |Jk | for all k ∈ Fα and aα ≤ |Fα |. Then for all α ∈ A, the measure algebra of aα · {−1, 1}Iα is isomorphic to a measure {−1, 1}Jk and hence, sub-algebra of k∈Fα Lp,∞   → Lp,∞  {−1, 1}Jk  c aα · {−1, 1}Iα α∈A k∈Fα α∈A (4.1.6) {−1, 1}Jn ∼ Lp,∞ . n∈N aα = ∞, there is a finite subset G1 ⊂ Aα1 such that On the other hand, since α∈Aα1 aα ≥ 1. Note that for all α ∈ G1 , |Iα | ≥ |J1 |. Then the measure algebra of {−1, 1}J1 α∈G1 aα · {−1, 1}Iα . Since for each n = 2, 3, . . . , is isomorphic to a measure sub-algebra of α∈G1 n−1 aα = ∞, there is a finite subset Gn ⊂ Aαn \ α∈Aαn Gk aα ≥ 1. such that α∈Gn k=1 Since for all n = 2, 3, . . . , |Iα | ≥ |Jn | for all α ∈ Gn , the measure algebra of {−1, 1}Jn is aα · {−1, 1}Iα and hence, isomorphic to a measure sub-algebra of α∈Gn {−1, 1}Jn Lp,∞ c → Lp,∞ aα · {−1, 1}Iα n∈N α∈Gn n∈N c → Lp,∞ (4.1.7) aα · {−1, 1}Iα . α∈A By Theorem 1.3, the embeddings (4.1.6) and (4.1.7) implies Lp,∞ aα · {−1, 1}Iα α∈A ∼ Lp,∞ {−1, 1}Jn . n∈N The following lemma is a result from set theory. This will be used to prove the next proposition. 4.1 The Classification 65 Lemma 4.3. Let A be a well ordered set of ordinal τ . Suppose Aα = {γ ∈ A : γ ≥ α} is uncountable for all α ∈ A. (4.1.8) If for every α ∈ A, we assign a non-zero countable ordinal βα , then βα ≤ τ. (4.1.9) α[...]... non-atomic weak Lp spaces in a ’standard form’ This will be done in Chapter 3, where we also observe necessary conditions of isomorphism of purely non-atomic weak Lp spaces and characteristic of embeddings between certain purely non-atomic weak Lp spaces Based on this ’standard’ representation and the results in Chapter 3, we are able to obtain an isomorphic classification of purely non-atomic weak Lp spaces. .. both Lp, ∞ [0, 1] and Lp, ∞ [0, ∞) are isomorphic to p,∞ He extended this result for purely atomic measure spaces The isomorphic classification of purely atomic weak Lp spaces was done in [8] Leung also studied the isomorphic relationship between purely atomic and purely nonatomic weak Lp spaces In [9], he showed that the isomorphism of atomic and non-atomic 1.2 The Objectives and Outline of the Thesis weak. .. classification of purely non-atomic weak Lp spaces This is the main objective of the thesis Its content is divided into four chapters, including the introduction chapter The investigation starts with the study of subspace structure of Lp, ∞ ({−1, 1}I ) in Chapter 2, where we try to find analogy for the results in [7] Then we continue to study the embeddings between purely non-atomic weak Lp spaces In doing so, we... Haar system) {hn }∞ are elements of Lp [0, 1] In n=1 fact, they form an unconditional basis of Lp [0, 1] for 1 < p < ∞ ([11], Theorem 2.c.6) The following theorem observes a property of the Haar functions as elements of weak Lp space Theorem 1.7 ([9, Proposition 8]) Let (hn )∞ be the Haar functions on [0, 1] and n=1 (Ω, Σ, µ) be a measure space If T : M 2,∞ [0, 1] → Lp, ∞ (Ω, Σ, µ) is an embedding, then... determine the uniqueness of the classification has only been partially solved We will present both the isomorphic classification and partial results on its uniqueness in Chapter 4 8 Chapter 2 Subspace Structure of Lp, ∞ {−1, 1}I In [7], Leung showed that p,∞ c ∼ Lp, ∞ [0, 1] ∼ Lp, ∞ [0, ∞) Obviously, Lp, ∞ [0, 1] → Lp, ∞ [0, ∞) and prior to this, Leung [5] had already proven p,∞ c → Lp, ∞ [0, 1] (2.0.1) Hence,... f1 , , fn in Lp, ∞ [0, 1] 1.2 The Objectives and Outline of the Thesis Carothers and Dilworth, [1], [2], studied the subspace structure on Lorentz spaces Lp, q [0, ∞) for q < ∞ In particular, they showed that for 1 < p < ∞, 1 ≤ q < ∞, and p = q, Lp, q [0, 1] and Lp, q [0, ∞) are not isomorphic ([1, Corollary 3.2]) The subspace structure of Lp, ∞ [0, 1] and p,∞ was studied by Leung in [5] and [6] In the... from Lp, ∞ Ji → Lp, ∞ ({−1, 1}I ) for 1 < p < 2 Proposition 2.11 Let 1 < p < 2 Define a mapping T : Lp, ∞ Ji → Lp, ∞ ({−1, 1}I ) by fi T i∈I ˜ fi , := w − i∈I ˜ where fi is defined in (2.1.6) Then T is an embedding Proof Let fi ∈ Lp, ∞ Ji First, we show that w − ˜ fi converges in Lp, ∞ ({−1, 1}I ) i∈I ˜ By Lemma 2.6, (fi )i∈I are symmetric random variables on {−1, 1}I Then for any finite 2.1 Embeddings from Lp, ∞... 10 i∈I p,∞ (Γ) observe that if there is an embedding from into Lp, ∞ ({−1, 1}I ) for 1 < p < 2, then the images of the unit vectors under this embedding are not independent 2.1 Embeddings from Lp, ∞ into Lp, ∞ {−1, 1}I Ji i∈I In this section we will prove the following Theorem 2.1 If 1 < p < 2, then Lp, ∞ c → Lp, ∞ {−1, 1}I Ji i∈I For any f ∈ Lp, ∞ fi , where fi = f |Ji Theorem 2.1 Ji , we write f = i∈I... from Lp, ∞ Ji into Lp, ∞ {−1, 1}I 16 i∈I The next three lemmas lead to Proposition 2.10, which shows that for any finite ˜ fi subset F of I, 1/2 2 and i∈F simple fact of an averaging operator fi are i∈F on Lp, ∞ [0, 1] equivalent We begin by recalling a Lemma 2.7 Let 1 < p < ∞ and n ∈ N Define an operator Pn on Lp, ∞ [0, 1] by 2n k/2n 2n Pn f = (k−1)/2n k=1 Then Pn f f dλ 1[(k−1)2−n ,k2−n ) f for all f ∈ Lp, ∞... (aj ) is the decreasing re|F |2n arrangement of (aj )j=1 And thus, |F |2n 2n ai,k 2 |F |2n = aj 2 j=1 i∈F k=1 j=1 fi ∈ Lp, ∞ ( Lemma 2.9 Let 1 < p < 2, i∈F 2 2/p Ji ), and F be a finite subset of I Then 1/2 1/2 ˜ fi 2n j ˜ fi sup n∈N 2 · 1Ωn i∈F fi = 1 Let i ∈ I and n ∈ N If ω ∈ Ωn , Proof Without loss of generality, assume i∈F 2.1 Embeddings from Lp, ∞ into Lp, ∞ {−1, 1}I Ji 18 i∈I then 2n ˜ fi (ω) = . ISOMORPHIC CLASSIFICATION OF WEAK L P SPACES RUDY SABARUDIN (M.Sc., National University of Singapore) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPART MENT OF MATHEMATICS NAT. . . . . . . . . 45 3.3 Characterization of An Embedding Between Weak L p Spaces, p ≥ 2 . . . 52 4 Isomorphic Classification of Purely Non-Atomic Weak L p Spaces 57 4.1 The Classification . . . study of isomorphic problems of weak L p spaces and state the objectives and outline of the thesis. In Chapter 2, Leung’s result in [7] will be extended for some purely non-atomic measure spaces.