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

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 K script upper K -theory of interpolation. As applications, we obtain Orlicz space mapping properties for the Hardy–Littlewood maximal function, approximate identities, and Calderón–Zygmund singular integral operators. These mapping properties are shown to be optimal for the Hardy–Littlewood maximal function and the approximate identities.