Optimal estimates for the Hardy averaging operator

✍ Scribed by Aleš Nekvinda; Luboš Pick

• Publisher: John Wiley and Sons
• Year: 2010
• Tongue: English
• Weight: 135 KB
• Volume: 283
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200711155

✦ Synopsis

Abstract

Let be the one‐dimensional Hardy averaging operator. It is well‐known that A is bounded on L^p^ whenever 1 < p ≤ ∞. We improve this result in the following sense: we introduce a pair of new function spaces, the ‘source’ space S~p~, which is strictly larger than L^p^, and the ‘target’ space T~p~, which is strictly smaller than L^p^, and prove that A is bounded from S~p~ into T~p~. Moreover, we show that this result cannot be improved within the environment of solid Banach spaces. We present applications of this result to variable‐exponent Lebesgue spaces L^p (x)^ (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)