# Duals of Optimal Spaces for the Hardy Averaging Operator

### Aleš Nekvinda

Czech Technical University, Praha, Czech Republic### Luboš Pick

Charles University, Praha, Czech Republic

## Abstract

The Hardy averaging operator $Af(x):=x1 ∫_{0}f(t)dt$ is known to map boundedly the 'source' space $S_{p}$ of functions on $(0,1)$ with finite integral

\[ \int_0^1 \esup_{t\in(x,1)}\frac1{t}\int_0^t |f|^p dx \]into the `target' space $T_{p}$ of functions on $(0,1)$ with finite integral

\[ \int_0^1 \esup_{t\in(x,1)}|f(t)|^p dx \]whenever $1<p<∞$. Moreover, the spaces $S_{p}$ and $T_{p}$ are optimal within the fairly general context of all Banach lattices. We prove a duality relation between such spaces. We in fact work with certain (more general) weighted modifications of these spaces. We prove optimality results for the action of $A$ on such spaces and point out some applications to the variable-exponent spaces. Our method of proof of the main duality result is based on certain discretization technique which leads to a~discretized characterization of the optimal spaces.

## Cite this article

Aleš Nekvinda, Luboš Pick, Duals of Optimal Spaces for the Hardy Averaging Operator. Z. Anal. Anwend. 30 (2011), no. 4, pp. 435–456

DOI 10.4171/ZAA/1443