A version of Gehring lemma in Orlicz spaces

Abstract

We present a version of the Gehring lemma, showing higher integrability in the scale of Orlicz spaces for a function $g$ satisfying reverse Hölder's inequalities of the type

under suitable integrability conditions on $f$ which do not imply boundedness. We describe explicitly in the general case how the improved integrability of $g$ depends on the assumptions on $f$, thus extending results of [4, 2] which deal with $f$ exponentially integrable.

We also present some applications of our result to the theory of mappings of finite inner distortion.