The Hardy averaging operator between Orlicz spaces and a new gauge functional characterizing a given Orlicz space

Abstract

We prove that the largest Orlicz space into which the Hardy averaging operator maps coincides with the largest rearrangement-invariant (r.i.) space mapping into that space; in other words, the optimal Orlicz domain is automatically the optimal r.i.\ domain. More generally, we show that every Luxemburg--Orlicz norm is equivalent to a sublinear functional which makes more computable a certain expression for a dual norm arising in the $\K$-theory of interpolation. As applications we obtain Orlicz-space mapping properties for the Hardy--Littlewood maximal function, approximate identities and Calder\'on--Zygmund singular integral operators. These mapping properties are shown to be optimal for the Hardy-Littlewood maximal function and the approximate identities.