Isomorphic classification of Lp,q-spaces, II

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Abstract

This is a continuation of the papers [36] and [54], in which the isomorphic classification of Lp,q, for 1<p<, 1q<, pq, on resonant measure spaces, has been obtained. The aim of this paper is to give a complete isomorphic classification of Lp,q-spaces on general σ-finite measure spaces. Towards this end, several new subspaces of Lp,q(0,1) and Lp,q(0,) are identified and studied.

Introduction

This paper is devoted to the isomorphic classification of Lorentz spaces Lp,q on general σ-finite measure spaces, where 1<p<, 1q<, pq, which is a continuation of previous studies on the isomorphic classification of Lp,q on resonant measure spaces contained in [36], [54] and is motivated by earlier work on subspaces of Lp,q due to Carothers and Dilworth (see e.g. [12], [13], [14], [15], [19]). For the isomorphic classification of Lp-spaces (i.e., pn, n=1,2,, p, Lp(0,1)), we refer to [60, Part III] and [7, Chapter XII]. For the isomorphic classification of weak Lp-spaces, we refer to [37], [38], [39], [40].

The Lorentz spaces Lp,q were introduced by G.G. Lorentz in [46], [47], and their importance is demonstrated in several areas of analysis such as harmonic analysis, interpolation theory, etc. (see e.g. [8], [19] and references therein). Recall that if (Ω,Σ,μ) is a measure space, then for 1<p< and 1q<, the Lorentz space Lp,q(Ω) is the collection of all measurable functions f on Ω such thatfp,q:=(0f(t)qdtq/p)1/q<, where f denotes the decreasing rearrangement of |f| (see the next section). It is well-known that Lp,q(Ω) is separable when Ω is σ-finite. In the special case when (Ω,Σ,μ) coincides with N equipped with the counting measure, the space Lp,q(Ω) coincides with the familiar symmetric sequence space p,q [19], [42]. It is well-known that if 1qp<, then p,q is a norm, and if 1<p<q<, then it is a complete quasi-norm which is equivalent to a complete norm [8], [19]. It is clear that Lp,p(Ω) is the Lebesgue space Lp(Ω), and that it is important to note that Lp,q-spaces arise in the Lions–Peetre K-method of interpolation [12], [19].

It is known that Lp,q(0,1) and Lp,q(0,) are not isomorphic to each other for any 1<p<, 1q<, pq, and p,q does not embed into Lp,q(0,1) as a complemented subspace [13], [19]. A stronger result showing that p,q does not embed isomorphically into Lp,q(0,1) was proved in [36] and [54] using the so-called “subsequence splitting lemma” introduced and studied in [56] (see also [5], [21], [23], [24], [26], [55]). It is interesting to note that there are symmetric function spaces on (0,1) which contain isomorphic p,q-copies, 1<p<2, 1q< [51]. It is also known [36], [54] that Lp,q(0,) does not embed isomorphically into the space Lp,q(0,1)p,q. Hence, if 1<p<, 1q<, pq, thenp,qn,n=1,2,,p,q,Lp,q(0,1),Lp,q(0,1)p,qandLp,q(0,) is the full list of pairwise non-isomorphic Lp,q-spaces over a resonant measure space.

In the present paper, we extend results in [36], [54] to the setting of general σ-finite measure spaces. Below, we briefly introduce the structure of the present paper. Unless stated otherwise, we will always assume that 1<p<, 1q< and pq.

In Section 3, we prove some results on the finite representability of p,qn, which are important auxiliary tools. In particular, using a type/cotype argument and Kadec–Pełczyński theorem, we show that p,qn does not embed uniformly into q when 1<p<min{2,q} or p>max{2,q} or q>2 (see Corollary 3.3).

It is well-known that any σ-finite measure space is the direct sum of an atomless measure space and an atomic measure space. Recall that all separable infinite (or finite) atomless measure spaces are isomorphic to each other [11, Theorem 9.3.4], and Lp,q(Ω) is isomorphic to p,qn, 1n, if Ω is an atomic resonant measure space. The main concern of this paper is the case when the measures of atoms are not necessarily the same in an atomic measure space. In Section 4, we classify infinite-dimensional Lp,q-spaces on different atomic measure spaces and show that any Lp,q-space on such a measure space is isomorphic to a direct sum of Lp,q-spaces of the following types (see below for definitions and Section 4)p,q,1,p,q,0(I),p,q,0(F),and p,q,. In Table 1, we present a complete isomorphic classification of these spaces. Note that all Banach spaces X of type p,q,1 are isomorphic to p,q.

If the atoms An in Ω satisfy the conditions that μ(An)0 and μ(An)=, then the Lp,q-space on such a measure space is said to be of type p,q,0(I). Johnson et al. [28, p. 31] introduced the subspace UY of an arbitrary symmetric function space Y on [0,), which is spanned by the characteristic functions of An. Such spaces are natural generalizations of the space Xp introduced by Rosenthal [53]. For any fixed symmetric function space Y on [0,), it is proved in [28, Theorem 8.7] (see also [43, Proposition 2.f.7]) that UY does not depend on the particular sequence {An} used (up to an isomorphism). When Y=Lp,q, the space UY coincides with an Lp,q-space of type p,q,0(I) and, for simplicity, we denote Up,q:=UY=ULp,q. By using techniques different those in [36], [54], we show that Up,q does not embed into p,q (indeed, it does not embed into p,qLp,q(0,1), see Theorem 5.7 below), which is a far-reaching generalization of [36, Theorem 11] and [54, Theorem 10]. Another interesting case is when the atoms An satisfy the condition n=1μ(An)< (or μ(An)n). Any Lp,q-space on a measure space of such type is said to be of type p,q,0(F) (resp. p,q,). It is well known that for any disjointly supported sequence of unit vectors in Lp,q(0,1), there exists a subsequence equivalent to the unit vector basis of q [19, Theorem 5] (see also [13]). We establish a quantitative version of this result (see Proposition 4.10, Proposition 4.25), giving a criterion for the normalized characteristic functions of atoms (in the setting of p,q,0(F) and p,q,) to be equivalent to the unit vector basis of q. We also show that (n=1p,qn)q is isomorphic to a space of the type p,q,0(F) (or p,q,) for some particular choice of {An}. In particular, we obtain that (n=1p,qn)q is a subspace of Lp,q(0,1).

In Section 5, we consider Lp,q-spaces on arbitrary σ-finite measure spaces with non-trivial atomless part. Up to isomorphism, any such Lp,q-space is one of the followingLp,q(0,1),Lp,q(0,1)p,q,1,Lp,q(0,1)p,q,0(I),Lp,q(0,1)p,q,,Lp,q(0,). We give a full characterization of the isomorphic embeddings between Lp,q-spaces of all types listed above. It is clear that any space of the first four types is a complemented subspace of Lp,q(0,). We show that Lp,q(0,) does not embed into any space of the first four types, which extends [36, Theorem 11] and [54, Theorem 10] significantly. We also establish an Lp,q-space version of a Theorem for Orlicz sequence spaces due to Lindenstrauss and Tzafriri [43, Theorem 2.c.14] (see [7, Ch. XII, Theorem 9] for the case of Lp-spaces), showing that there exists an isomorphic embeddingT:Lp,q(0,1)Up,q if and only if p=q=2. In the language of graph theory, the tree below is the Hasse diagram for the partially ordered set consisting of the equivalence classes of Lp,q(Ω) under Banach isomorphism with the order relation. For any spaces XY listed in the tree following, X is isomorphic to a subspace (indeed, a complemented subspace) of Y if and only if X can be joined to Y through a descending branch (see Table 2).

Our notations and terminology are standard and all unexplained terms may be found in [1], [28], [42], [43].

We would like to thank Professor J. Arazy for helpful discussions concerning results presented in his paper [3], which contains useful techniques in the study of isomorphic embedding of Banach spaces. We also thank Professor W.B. Johnson for his comments on finite representability of p-spaces and his help in proving Proposition 3.2. We thank A. Kuryakov for many joint discussions, and T. Scheckter for his careful reading of this paper, and Professor E. Semenov for useful comments and sharing the paper [10], and we thank D. Zanin for helpful discussions. The second author was supported by the Australian Research Council (FL170100052). Authors thank the anonymous referee for reading the paper carefully and providing thoughtful comments, which improved the exposition of the paper.

Section snippets

Preliminaries

Let (Ω,Σ,μ) be a measure space with a σ-finite measure μ, defined on σ-algebra Σ, and let L(Ω) be the algebra of all classes of equivalent measurable real-valued functions on (Ω,Σ,μ). For any function fL(Ω), its distribution df(s) is given bydf(s):=μ({f>s}),s>0. Denote by L0(Ω) the subalgebra of L(Ω) consisting of all functions f such that d|f|(s)< for some s>0. For every fL0(Ω), its non-increasing rearrangement is defined byf(t):=inf{s>0:d|f|(s)t},>0. In the special case when the measure

Some results on the uniform embedding of p,qn

Recall that p,qn is said to embed uniformly in a Banach space X if for every nN, there exists an operator Tn:p,qnX such that supnTnT1<. The connection between type/cotype and the uniform embedding of pn into a Banach space has been widely studied (see [50] and [49, Theorem 3.5]).

It is well-known that p↪̸Lp,q(0,) [13], [19]. However, p is finitely representable in p,q, and therefore, in Lp,q(0,). In particular, Carothers and Flinn [15] obtained a quantitative result in the sense

Isomorphic classification of Lp,q: the atomic case

This section contains main results of the present paper. Throughout this section, we always assume that1<p<,1q<,pq, and (Ω,Σ,μ) is an atomic measure space and such that Ω consists of atoms An, 1n<, with μ(An)<. We denote Lp,q(Ω) by p,q({An}) and the characteristic functions in Lp,q(Ω) generated by An are denoted by χAn.

Isomorphic classification of Lp,q: general σ-finite case with a non-trivial atomless part

In this section, we consider Lp,q-spaces on a general σ-finite measure space with a non-trivial atomless part.

We state below results of isomorphic classification of Lp,q-spaces obtained in this section and supply the proof which is also based on a number of results given later in this subsection.

Theorem 5.1

Let 1<p<, 1q<, pq. Let (Ω,Σ,μ) be a σ-finite measure space which is not purely atomic. Then, Lp,q(Ω) is of one of the following types (up to an isomorphism)Lp,q(0,1),Lp,q(0,),Lp,q(0,1)Up,q,Lp,q(0,

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