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

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