The sharp reverse Hardy inequality in BMO for nonincreasing functions

Abstract

Let $Hf(x)=x^{-1}\int_0^x f(t)\,dt$ be the Hardy operator on $\mathbb R_+$. Korenovskii proved that $$ \Vert Hf\Vert_{\mathrm{BMO}}\geq \frac{eα_0}{4}\Vert f\Vert_{\mathrm{BMO}}, $$ for every nonincreasing and locally integrable $f$, where $α_0$ is defined by the relation $\Vert Hχ_{(0,1)}\Vert_{\mathrm{BMO}} = α_0 \Vert χ_{(0,1)}\Vert_{\mathrm{BMO}}$, and conjectured that the factor $e/4$ could be removed. We prove this conjecture by showing that every nonincreasing locally integrable $f$ satisfies $$ \Vert Hf\Vert_{\mathrm{BMO}}\geq α_0\Vert f\Vert_{\mathrm{BMO}}. $$ The constant $α_0$ is optimal, with equality for the one-jump functions $χ_{(0,a)}$.