Grand Lebesgue spaces with respect to measurable functions

Abstract

Let 1<p<∞. Given Ω⊂Rn a measurable set of finite Lebesgue measure, the norm of the grand Lebesgue spaces Lp)(Ω) is given by |f|Lp)(Ω)=sup0<ε<p−1ε1p−ε(1|Ω|∫Ω|f|p−εdx)1p−ε. In this paper we consider the norm |f|Lp),δ(Ω) obtained replacing ε1p−ε by a generic nonnegative measurable function δ(ε). We find necessary and sufficient conditions on δ in order to get a functional equivalent to a Banach function norm, and we determine the “interesting” class Bp of functions δ, with the property that every generalized function norm is equivalent to a function norm built with δ∈Bp. We then define the Lp),δ(Ω) spaces, prove some embedding results and conclude with the proof of the generalized Hardy inequality.