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LITTLEWOOD–PALEY CHARACTERIZATION OF WEIGHTED HARDY SPACES ASSOCIATED WITH OPERATORS

Part of: Harmonic analysis in several variables Special classes of linear operators

Published online by Cambridge University Press:  10 November 2016

GUORONG HU
GUORONG HU*
Affiliation:
Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, PR China email [email protected], [email protected]
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Abstract

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Let $(X,d,\unicode[STIX]{x1D707})$ be a metric measure space endowed with a distance $d$ and a nonnegative, Borel, doubling measure $\unicode[STIX]{x1D707}$. Let $L$ be a nonnegative self-adjoint operator on $L^{2}(X)$. Assume that the (heat) kernel associated to the semigroup $e^{-tL}$ satisfies a Gaussian upper bound. In this paper, we prove that for any $p\in (0,\infty )$ and $w\in A_{\infty }$, the weighted Hardy space $H_{L,S,w}^{p}(X)$ associated with $L$ in terms of the Lusin (area) function and the weighted Hardy space $H_{L,G,w}^{p}(X)$ associated with $L$ in terms of the Littlewood–Paley function coincide and their norms are equivalent. This improves a recent result of Duong et al. [‘A Littlewood–Paley type decomposition and weighted Hardy spaces associated with operators’, J. Geom. Anal.26 (2016), 1617–1646], who proved that $H_{L,S,w}^{p}(X)=H_{L,G,w}^{p}(X)$ for $p\in (0,1]$ and $w\in A_{\infty }$ by imposing an extra assumption of a Moser-type boundedness condition on $L$. Our result is new even in the unweighted setting, that is, when $w\equiv 1$.

Keywords

Hardy spacesweightsnonnegative self-adjoint operatorsheat semigroupLusin and Littlewood–Paley functionsspace of homogeneous type

MSC classification

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B30: $H^p$-spaces 42B35: Function spaces arising in harmonic analysis 47B25: Symmetric and selfadjoint operators (unbounded)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 103 , Issue 2 , October 2017 , pp. 250 - 267
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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