A Littlewood–Paley Type Decomposition and Weighted Hardy Spaces Associated with Operators

Abstract

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 second-order non-negative self-adjoint operator on $L^2(X)$. Assume that the semigroup $e^{-tL}$ generated by $L$ satisfies Gaussian upper bounds. In this article we establish a discrete characterization of weighted Hardy spaces $H_{L, S, w}^{p}(X)$ associated with $L$ in terms of the area function characterization, and prove its weighted atomic decomposition, where $0<p\le 1$ and a weight $w$ is in the Muckenhoupt class $A_{\infty }$. Further, we introduce a Moser type estimate for $L$ to show the discrete characterization for the weighted Hardy spaces $H_{L, G, w}^{p}(X)$ associated with $L$ in terms of the Littlewood–Paley function and obtain the equivalence between the weighted Hardy spaces in terms of the Littlewood–Paley function and area function.