# 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 $Af(x):=∣B(0,∣x∣)∣1 ∫_{B(0,∣x∣)}f(t)dt$ be the $n$-dimensional Hardy averaging operator. It is well known that $A$ is bounded on $L_{p}(Ω)$ with an open set $Ω⊂R_{n}$ whenever $1<p≤∞$. 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 $L_{p(⋅)}(Ω)$, 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 $Ω$ 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