Averaging operators on decreasing or positive functions: equivalence and optimal bounds (Q1801179)

The Hardy operator $S$ of a positive measurable function $f$ on $(0,\infty)$ is defined at point $x$ by the integral mean of $f(t)$ on the interval $[0,x]$, while its dual $S^*$ is the improper integral of $f(t)/t$ on $[x,\infty)$.\par The authors study optimal bounds for $S$-$I$ and $S^*$ (where $I$ is the identity operator) on the full range $1\le p\le\infty$ of $L^p$ functions for the case of decreasing, positive or general functions. There are many results, we state for example the following \begin{itemize} \item[1)] if $f$ is decreasing, then $\Vert Sf-f\Vert_p\le\frac{1}{p-1}\Vert f\Vert_p$ for $1<p<2$; \item[2)] if $2<p<\infty$ and $f$ is positive then $\Vert Sf-f\Vert_p\le\Vert f\Vert_p$; \item[3)] let $p\ge 2$ be a positive integer. Then for decreasing $f$, one has $\Vert Sf\Vert_p\le (p'/p)^{1/p}\cdot\Vert S^* f\Vert_p$, where $p'=p/(p-1)$. \end{itemize} All inequalities are sharp.