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LITTLEWOOD–PALEY CHARACTERIZATION OF WEIGHTED HARDY SPACES ASSOCIATED WITH OPERATORS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 103 / Issue 2 / October 2017
- Published online by Cambridge University Press:
- 10 November 2016, pp. 250-267
- Print publication:
- October 2017
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- Article
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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 
$w\equiv 1$.
