Sharp estimates for the identity minus Hardy operator on the cone of decreasing functions

Abstract

It is shown that if we restrict the identity minus Hardy operator on the cone of nonnegative decreasing functions $f$ in $L^{p}$, then we have the sharp estimate \begin{equation*} \left \| (I-H)f\right \| _{L^p}\leq \frac {1}{(p-1)^{\frac {1}{p}}}\left \| f\right \| _{L^p} \end{equation*} for $p=2,3,4,....$ In other words, \begin{equation*} \left \| f^{**}-f^* \right \| _{L^p}\leq \frac {1}{(p-1)^{\frac {1}{p}}} \left \| f\right \| _{L^p} \end{equation*} for each $f \in L^p$ and each integer $p\ge 2$. It is also shown, via a connection between the operator $I-H$ and Laguerre functions, that \begin{equation*} \|(1-\alpha ) I+\alpha (I-H)\|_{L^2\to L^2}=\|I-\alpha H\|_{L^2\to L^2}=1 \end{equation*} for all $\alpha \in [0,1]$.