Maximal inequalities for a best approximation operator in Orlicz spaces

Abstract

In this paper we study a maximal operator Mf related with the best ϕ approximation by constants for a function f ∈ L ϕ 0 loc(ℝn), where we denote by ϕ 0 for the derivative function of the C1 convex function ϕ. We get a necessary and sufficient condition which assure strong inequalities of the type R ℝn θ(M|f|) dx ¬ K R ℝn θ(|f|) dx, where K is a constant independent of f. Some pointwise and mean convergence results are obtained. In the particular case ϕ(t) = t p+1 we obtain several equivalent conditions on the functions θ that assures strong inequalities of this type.