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A maximal function characterization of the Hardy space for the Gauss measure

Authors: Giancarlo Mauceri, Stefano Meda and Peter Sjögren
Journal: Proc. Amer. Math. Soc. 141 (2013), 1679-1692
MSC (2010): Primary 42B30, 42B35; Secondary 42C10
Published electronically: November 2, 2012
MathSciNet review: 3020855
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Abstract: An atomic Hardy space $H^1(\gamma )$ associated to the Gauss measure $\gamma$ in $\mathbb {R}^n$ has been introduced by the first two authors. We first prove that it is equivalent to use $(1,r)$- or $(1,\infty )$-atoms to define this $H^1(\gamma )$. For $n=1$, a maximal function characterization of $H^1(\gamma )$ is found. In arbitrary dimension, we give a description of the nonnegative functions in $H^1(\gamma )$ and use it to prove that $L^p(\gamma )\subset H^1(\gamma )$ for $1<p\le \infty$.

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Additional Information

Giancarlo Mauceri
Affiliation: Dipartimento di Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova, Italia

Stefano Meda
Affiliation: Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, via R. Cozzi 53, 20125 Milano, Italy

Peter Sjögren
Affiliation: Mathematical Sciences, University of Gothenburg, Box 100, S-405 30 Gothenburg, Sweden — and — Mathematical Sciences, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden

Keywords: Gaussian measure, Gaussian Hardy space, maximal function, atomic decomposition
Received by editor(s): February 10, 2011
Received by editor(s) in revised form: September 7, 2011
Published electronically: November 2, 2012
Additional Notes: This work was partially supported by PRIN 2009 “Analisi Armonica”.
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.