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

We give a sharp sufficient condition on the distribution function, |{x∈Ω : p(x)≤ 1+λ}|, λ>0, of the exponent function p(·): Ω→ [1,∞) that implies the embedding of the variable Lebesgue space L^p(·)(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(·)((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 .