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

Mauceri, Giancarlo; Meda, Stefano; Sjögren, Peter

doi:10.1090/S0002-9939-2012-11443-1

A maximal function characterization of the Hardy space for the Gauss measure
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by Giancarlo Mauceri, Stefano Meda and Peter Sjögren PDF

Proc. Amer. Math. Soc. 141 (2013), 1679-1692 Request permission

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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