On the embedding between the variable Lebesgue space Lp(·)(Ω) and the Orlicz space L(log L)α(Ω)

We give a sharp sufficient condition on the distribution function, |∈ Ω :\,p(x)≤ 1+λ\|, λ>0, of the exponent function p(·): Ω → [1,∞) that implies the embedding of the variable Lebesgue space Lp(·)(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 Lp(·)((0,1)ⁿ), $n>1$. We then consider the integrability of the maximal function on variable Lebesgue spaces, where the exponent function p(·) approaches $1$ in value on some part of the domain. This result is an improvement of the result in~{CUF2}.