Averaging operators on decreasing or positive functions: Equivalence and optimal bounds

Abstract

We study the optimal bounds for the Hardy operator S minus the identity, as well as S and its dual operator S∗, on the full range 1 ≤ p ≤ ∞, for the cases of decreasing, positive or general functions (in fact, these two kinds of inequalities are equivalent for the appropriate cone of functions). For 1 < p ≤ 2, we prove that all these estimates are the same, but for 2 < p < ∞, they exhibit a completely different behavior.