LITTLEWOOD–PALEY CHARACTERIZATION OF WEIGHTED HARDY SPACES ASSOCIATED WITH OPERATORS

Abstract

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 Duonget 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$w\equiv 1$.