On the optimal relationships between [formula omitted]-norms for the Hardy operator and its dual for decreasing functions

Abstract

We prove sharp inequalities between Lp−norms (1<p<∞) of functions Hf and H∗f, where H is the Hardy operator, H∗ is its dual, and f is a nonnegative nonincreasing function on (0,∞). In particular, we extend one result obtained for integer p≥2 by Boza and Soria (2019), to the whole range of values p≥2.