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 is known to map boundedly the 'source' space of functions on with finite integral

\[ \int_0^1 \esup_{t\in(x,1)}\frac1{t}\int_0^t |f|^p dx \]

into the `target' space of functions on with finite integral

\[ \int_0^1 \esup_{t\in(x,1)}|f(t)|^p dx \]

whenever . Moreover, the spaces and 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 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