# Hardy Averaging Operator on Generalized Banach Function Spaces and Duality

### Yoshihiro Mizuta

Hiroshima Institute of Technology, Japan### Aleš Nekvinda

Czech Technical University, Praha, Czech Republic### Tetsu Shimomura

Hiroshima University, Graduate School of Education, Higashi-Hiroshima, Japan

## 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 $L\sp p(\Omega)$ with an open set $\Omega \subset \R^n$ whenever $1<p\leq\infty$. 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 $A'$ 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.

## Cite this article

Yoshihiro Mizuta, Aleš Nekvinda, Tetsu Shimomura, Hardy Averaging Operator on Generalized Banach Function Spaces and Duality. Z. Anal. Anwend. 32 (2013), no. 2, pp. 233–255

DOI 10.4171/ZAA/1483