Elsevier

Nonlinear Analysis

Volume 144, October 2016, Pages 23-31
Nonlinear Analysis

From Arzelà–Ascoli to Riesz–Kolmogorov

https://doi.org/10.1016/j.na.2016.06.004Get rights and content

Abstract

In this paper we study totally bounded sets in Banach function spaces (BFS), from which we characterize compact sets (via Hausdorff criterion) in some non-standard function spaces which fall under the umbrella of BFS. We obtain a Riesz–Kolmogorov compactness theorem for the grand variable exponent Lebesgue spaces.

Introduction

In the last decades, there has been increased interest in the so-called non-standard function spaces not only due to mathematical curiosity but also fueled by various applications. We mention only variable exponent Lebesgue and Morrey spaces as well as grand Lebesgue spaces. Function spaces with variable exponent are a very active area of research and one of the reasons is the wide variety of applications of such spaces, e.g., in the modeling of electrorheological fluids  [35] as well as thermo-rheological fluids  [4], in the study of image processing  [1], [40] and in differential equations with non-standard growth. Lebesgue spaces with variable exponent in the framework of quasi-metric measure space have also been studied by several authors, for which we refer to  [3], [13]. On the other hand, grand Lebesgue spaces have their genesis in the problem of the integrability of the Jacobian under minimal hypothesis  [21], but it was found that these are the right spaces in which some nonlinear equations in the theory of PDEs have to be considered  [10], [19], [25], [24], among other applications.

Some compactness criteria were studied in those particular non-standard function spaces, e.g.  [5], [17], [16], [26], [31], [33] but that problem can be put in a more general framework, namely in the case of Banach function spaces (BFS) as done in  [32]. In this paper, relying on the Arzelà–Ascoli theorem, we obtain some description of totally bounded sets in Banach function spaces on arbitrary metric measure spaces, which permits us to obtain compactness criteria for several function spaces via Hausdorff criterion.

In the classical Lp-spaces, the relatively compact sets are characterized by the celebrated Riesz–Kolmogorov theorem  [11], [28], [34], [36], [37], [38]. Let us mention about some generalizations of the Riesz–Kolmogorov theorem. For instance, the papers  [16], [29], [22], [18] contain the characterizations of precompact sets in Lp(X,ϱ,μ), where (X,ϱ,μ) is a metric measure space. Furthermore, Weil  [39] showed the compactness theorem in Lp(G), where G is a locally compact group. Pego  [30] (see also  [14], [15]) stated the Riesz–Kolmogorov theorem for p=2 in terms of the Fourier transform.

The remainder of the paper is structured as follows. In Section  2 we introduce the required notions related to Banach function spaces. We also recall basic facts about metric measure spaces there. Characterization of totally bounded sets in BFS on metric measure spaces is proved in Section  3. In Section  4 we obtain compactness results in the so-called grand variable exponent Lebesgue spaces.

Section snippets

Banach function spaces

We shall start from recalling some notation and basic facts about Banach function spaces. Most of the properties for these spaces can be found in the book by Edmunds and Evans  [9].

Definition 2.1

Let (Ω,μ) be a σ-finite, complete measure space. A normed space (E,E) with EL0(Ω,μ) is called a Banach function space if the following conditions are satisfied

  • (A1)

    if fE, then |f|E=fE;

  • (A2)

    if 0gf, then gEfE;

  • (A3)

    if 0fnf, then fnEfE;

  • (A4)

    if AΩ and μ(A)<, then χAE;

  • (A5)

    if AΩ and μ(A)<, then there exists a constant

Main results

The main result of this paper is the following.

Theorem 3.1

Assume that (X,ρ,μ) is a proper metric measure space with continuous measure. Let E=E(X,μ) be a Banach function space. If the family FE satisfies the following conditions:

  • (a)

    F is bounded in E, i.e.supfFfE<;

  • (b)

    limr0supfFB(,r)fdμfE=0;

  • (c)

    for some x0XlimRsupfFfχXB(x0,R)E=0,

then the family F is totally bounded in E.

Remark

It is well known that a doubling metric measure space is proper (i.e. every closed bounded set is compact) if and only if it is

Applications

In this section we want to discuss some particular case of BFS, the so-called grand variable exponent Lebesgue space. We first remember the so-called Hausdorff criterion for compactness which loosely states that a set is compact if and only if it is complete and totally bounded.

In  [23] the grand variable exponent Lebesgue spaces were introduced, which unify two non-standard function spaces: variable exponent Lebesgue spaces and grand Lebesgue spaces.

We start by defining the variable exponent

Acknowledgments

H. Rafeiro was partially supported by Pontificia Universidad Javeriana under the research project “Study of non-standard Banach spaces”, ID PPT: 6326. The authors also thank the anonymous referee for careful reading and very useful comments.

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