Hardy Averaging Operator on Generalized Banach Function Spaces and Duality

Abstract

Let $Af(x):=\frac{1}{|B(0,|x|)|}{\int }_{B(0,|x|)}f(t)dt$ be the $n$-dimensional Hardy averaging operator. It is well known that $A$ is bounded on $L^p(\Omega )$ with an open set $\Omega \subset {\mathbb{R}}^n$ whenever $1<p\le \infty$. We improve this result within the framework of generalized Banach function spaces. We in fact find the “source” space $S_X$, which is strictly larger than $X$, and the “target” space $T_X$, which is strictly smaller than $X$, under the assumption that the Hardy–Littlewood maximal operator $M$ is bounded from $X$ into $X$, and prove that $A$ is bounded from $S_X$ into $T_X$. We prove optimality results for the action of $A$ and its associate operator $A^{\prime }$ on such spaces and present applications of our results to variable Lebesgue spaces $L^{p(\cdot )}(\Omega )$, as an extension of A. Nekvinda and L. Pick [Math. Nachr. 283 (2010), 262–271; Z. Anal. Anwend. 30 (2011), 435–456] in the case when $n=1$ and $\Omega$ is a bounded interval.