Modular inequalities for the maximal operator in variable Lebesgue spaces

Abstract

A now classical result in the theory of variable Lebesgue spaces due to Lerner (2005) is that a modular inequality for the Hardy–Littlewood maximal function in Lp(⋅)(Rn) holds if and only if the exponent is constant. We generalize this result and give a new and simpler proof. We then find necessary and sufficient conditions for the validity of the weaker modular inequality ∫ΩMf(x)p(x)dx⩽c1∫Ω|f(x)|q(x)dx+c2,where c1,c2 are non-negative constants and Ω is any subset of Rn. As a corollary we get sufficient conditions for the modular inequality ∫Ω|Tf(x)|p(x)dx⩽c1∫Ω|f(x)|q(x)dx+c2,where T is any operator that is bounded on Lp(Ω), 1<p<∞.