On the embedding between the variable Lebesgue space $L^{p(\cdot)}(\Omega)$ and the Orlicz space $L(\log L)^{\alpha}(\Omega)$

TL;DR

This paper establishes a precise condition on the distribution of the variable exponent that guarantees the embedding of variable Lebesgue spaces into Orlicz spaces, with applications to differentiation of integrals and maximal functions.

Contribution

It provides a sharp sufficient condition for embedding variable Lebesgue spaces into Orlicz spaces, improving understanding of integrability and differentiation properties.

Findings

Derived a sharp condition on the distribution function for embedding

Established criteria for strong differentiation of integrals in variable Lebesgue spaces

Improved results on the integrability of maximal functions when the exponent approaches 1

Abstract

We give a sharp sufficient condition on the distribution function, $|\{x\in \Omega :\,p(x)\leq 1+\lambda\}|$, $\lambda>0$, of the exponent function $p(\cdot): \Omega \to [1,\infty)$ that implies the embedding of the variable Lebesgue space $L^{p(\cdot)}(\Omega)$ into the Orlicz space $L(\log L)^{\alpha}(\Omega)$, $\alpha>0$, where $\Omega$ is an open set with finite Lebesgue measure. As applications of our results, we first give conditions that imply the strong differentiation of integrals of functions in $L^{p(\cdot)}((0,1)^{n})$, $n>1$. We then consider the integrability of the maximal function on variable Lebesgue spaces, where the exponent function $p(\cdot)$ approaches $1$ in value on some part of the domain. This result is an improvement of the result in~\cite{CUF2}.