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Disjointly strictly singular inclusions between variable Lebesgue spaces

Francisco L. Hernández, César Ruiz, Mauro Sanchiz

Abstract

Disjointly strictly singular inclusions between variable Lebesgue spaces $L^{p(\cdot)}(μ)$ on finite measure are characterized. Suitable criteria in terms of the (bounded or unbounded) exponents are given. It is proved the equivalence of $L$-weak compactness (also called almost compactness) and disjoint strict singularity for variable Lebesgue space inclusions. For infinite measure any inclusion $L^{p(\cdot)}(μ) \hookrightarrow L^{q(\cdot)}(μ)$ is not disjointly strictly singular. No restrictions on the exponent are imposed.

Disjointly strictly singular inclusions between variable Lebesgue spaces

Abstract

Disjointly strictly singular inclusions between variable Lebesgue spaces on finite measure are characterized. Suitable criteria in terms of the (bounded or unbounded) exponents are given. It is proved the equivalence of -weak compactness (also called almost compactness) and disjoint strict singularity for variable Lebesgue space inclusions. For infinite measure any inclusion is not disjointly strictly singular. No restrictions on the exponent are imposed.
Paper Structure (6 sections, 24 theorems, 99 equations)

This paper contains 6 sections, 24 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Previous Results
  4. The finite measure case
  5. Regular exponents
  6. The infinite measure case

Key Result

Proposition 2.1

(Libro2 Thm 2.45, Libro Thm 3.3.1) Let $(\Omega,\mu)$ be an atomless infinite measure space and exponents $p(\cdot)$ and $q(\cdot)$. The inclusion ${L^{p(\cdot)}(\mu)} \hookrightarrow {L^{q(\cdot)}(\mu)}$ holds if and only if $q(\cdot)\leq p(\cdot)$$\mu$-a.e. and there exists $\lambda>1$ such tha where $\Omega_d=\{t\in\Omega : p(t)>q(t)\}$.

Theorems & Definitions (48)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 38 more