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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 30, Issue 4, 2011, pp. 435–456**

**DOI: 10.4171/ZAA/1443**

Published online: 2011-09-29

Duals of Optimal Spaces for the Hardy Averaging Operator

Aleš Nekvinda^{[1]}and Luboš Pick

^{[2]}(1) Czech Technical University, Praha, Czech Republic

(2) Charles University, Praha, Czech Republic

The Hardy averaging operator $Af(x):=\frac1x\int_0^x f(t)\,dt$ 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 $1< p < \infty$. 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.

*Keywords: *Hardy averaging operator, optimal target and domain spaces, associate spaces, discretization, Banach lattice, weights, weighted spaces, variable-exponent spaces

Nekvinda Aleš, Pick Luboš: Duals of Optimal Spaces for the Hardy Averaging Operator. *Z. Anal. Anwend.* 30 (2011), 435-456. doi: 10.4171/ZAA/1443