LITTLEWOOD-PALEY CHARACTERIZATION OF WEIGHTED HARDY SPACES ASSOCIATED WITH OPERATORS

Let (X, d, mu) be a metric measure space endowed with a distance d and a nonnegative, Borel, doubling measure mu. 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 is an element of (0, infinity) and w is an element of A(infinity), the weighted Hardy space H-L,H-S,H-w (p) (X) associated with L in terms of the Lusin (area) function and the weighted Hardy space H-L,H-G,H-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,H-S,H-w (p) (X) = H-L,H-G,H-w (p) (X) for p is an element of (0; 1] and w is an element of A(infinity) 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 equivalent to 1.