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

Abstract

We give a sharp sufficient condition on the distribution function, $|\{x\in \Omega :p(x)\le 1+\lambda \}|$, $\lambda >0$, 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}.