JournalszaaVol. 30, No. 4pp. 435–456

Duals of Optimal Spaces for the Hardy Averaging Operator

  • Aleš Nekvinda

    Czech Technical University, Praha, Czech Republic
  • Luboš Pick

    Charles University, Praha, Czech Republic
Duals of Optimal Spaces for the Hardy Averaging Operator cover
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Abstract

The Hardy averaging operator Af(x):=1x0xf(t)dtAf(x):=\frac1x\int_0^x f(t)\,dt is known to map boundedly the 'source' space SpS^p of functions on (0,1)(0,1) with finite integral

01\esupt(x,1)1t0tfpdx\int_0^1 \esup_{t\in(x,1)}\frac1{t}\int_0^t |f|^p dx

into the `target' space TpT^p of functions on (0,1)(0,1) with finite integral

01\esupt(x,1)f(t)pdx\int_0^1 \esup_{t\in(x,1)}|f(t)|^p dx

whenever 1<p<1< p < \infty. Moreover, the spaces SpS^p and TpT^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 AA 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