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The Journal of Geometric Analysis

, Volume 26, Issue 4, pp 2639–2663 | Cite as

Riesz Transforms Characterizations of Hardy Spaces \(H^1\) for the Rational Dunkl Setting and Multidimensional Bessel Operators

  • Jacek Dziubański
Article

Abstract

We characterize the Hardy space \(H^1\) in the rational Dunkl setting associated with the reflection group \(\mathbb {Z}_2^n\) by means of special Riesz transforms. As a corollary we obtain Riesz transforms characterization of \(H^1\) for product of Bessel operators in \((0,\infty )^n\).

Keywords

Dunkl theory Riesz transform Hardy space Maximal operator Atomic decomposition 

Mathematics Subject Classification

Primary: 42B30 Secondary: 33C52 35J05 42B25 42B35 42C05 

Notes

Acknowledgments

The author wishes to thank Jean-Philippe Anker, Paweł Głowacki and Rysiek Szwarc for their remarks. The author is greatly indebted to Bartosz Trojan for his suggestions which shortened the original proof of Theorem 1.3. Finally, the author wants to thank the referee for her/his careful reading of the manuscript and helpful comments which improved the presentation of the paper. Research supported by the Polish National Science Center (Narodowe Centrum Nauki, grant DEC-2012/05/B/ST1/00672) and by the University of Orléans.

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Copyright information

© Mathematica Josephina, Inc. 2015

Authors and Affiliations

  1. 1.Uniwersytet Wrocławski, Instytut MatematycznyWrocławPoland

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