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

Abstract

We give a sharp sufficient condition on the distribution function, $|\{x\in Ω:\,p(x)\leq 1+λ\}|$, $λ>0$, of the exponent function $p(\cdot): Ω\to [1,\infty)$ that implies the embedding of the variable Lebesgue space $L^{p(\cdot)}(Ω)$ into the Orlicz space $L(\log L)^α(Ω)$, $α>0$, where $Ω$ 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}.