Optimal estimates for the fractional Hardy operator
Volume 227 / 2015
Abstract
Let $A_{\alpha}f(x) = |B(0,|x|)|^{-\alpha/n} \int_{B(0,|x|)} f(t) \,dt$ be the $n$-dimensional fractional Hardy operator, where $0<\alpha \le n$. It is well-known that $A_{\alpha}$ is bounded from $L^p$ to $L^{p_\alpha}$ with $p_\alpha=np/(\alpha p-np+n)$ when $n(1-1/p)<\alpha \le n$. We improve this result within the framework of Banach function spaces, for instance, weighted Lebesgue spaces and Lorentz spaces. We in fact find a `source' space $S_{\alpha,Y}$, which is strictly larger than $X$, and a `target' space $T_Y$, which is strictly smaller than $Y$, under the assumption that $A_{\alpha}$ is bounded from $X$ into $Y$ and the Hardy–Littlewood maximal operator $M$ is bounded from $Y$ into $Y$, and prove that $A_{\alpha}$ is bounded from $S_{\alpha,Y}$ into $T_Y$. We prove optimality results for the action of $A_{\alpha}$ and the associate operator $A'_\alpha$ on such spaces, as an extension of the results of Mizuta et al. (2013) and Nekvinda and Pick (2011). We also study the duals of optimal spaces for $A_\alpha$.
