Duals of Optimal Spaces for the Hardy Averaging Operator

Abstract

The Hardy averaging operator $Af(x):=\frac{1}{x}{\int }_0^{x}f(t)dt$ is known to map boundedly the ‘source’ space $S^p$ of functions on $(0,1)$ with finite integral

into the ‘target’ space $T^p$ of functions on $(0,1)$ with finite integral

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.