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

Let (Formula presented.) be a metric measure space endowed with a distance (Formula presented.) and a nonnegative Borel doubling measure (Formula presented.). Let (Formula presented.) be a second-order non-negative self-adjoint operator on (Formula presented.). Assume that the semigroup (Formula presented.) generated by (Formula presented.) satisfies Gaussian upper bounds. In this article we establish a discrete characterization of weighted Hardy spaces (Formula presented.) associated with (Formula presented.) in terms of the area function characterization, and prove its weighted atomic decomposition, where (Formula presented.) and a weight (Formula presented.) is in the Muckenhoupt class (Formula presented.). Further, we introduce a Moser type estimate for (Formula presented.) to show the discrete characterization for the weighted Hardy spaces (Formula presented.) associated with (Formula presented.) 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.