Optimal relations between Lp-norms for the Hardy operator and its dual

Abstract

We obtain sharp two-sided inequalities between $L^p-$norms $(1<p<\infty )$ of functions $Hf$ and $H^{\ast }f$, where $H$ is the Hardy operator, $H^{\ast }$ is its dual, and $f$ is a nonnegative measurable function on $(0,\infty ).$ In an equivalent form, it gives sharp constants in the two-sided relations between $L^p$-norms of functions $H\f-\f$ and $\f$, where $\f$ is a nonnegative nonincreasing function on $(0,+\infty )$ with $\f(+\infty)=0.$ In particular, it provides an alternative proof of a result obtained by N. Kruglyak and E. Setterqvist (2008) for $p=2k(k\in \mathbb{N})$ and by S. Boza and J. Soria (2011) for all $p\ge 2$, and gives a sharp version of this result for $1<p<2$.