Maximal Inequalities in Orlicz Spaces

Abstract

Given non negative measurable real valued functions f and g, we get inequalities of the type ! Ω Ψ(f) dµ ≤ K ! Ω Ψ( g c ) dµ, assuming weak type inequalities µ({f>a}) ≤ K ! {f>a} ϕ( g a ) dµ where ϕ, ψ : R+ 0 → R+ 0 are nondecreasing functions related by ≺N and where Ψ is a Young function given by Ψ(x) = ! x 0 ψ(t) dt. We apply these results to best approximation operators and sub additive operators.