Hardy Spaces Associated with Non-negative Self-adjoint Operators and Ball Quasi-Banach Function Spaces on Doubling Metric Measure Spaces and Their Applications

Let (𝒳,d,μ ) be a doubling metric measure space in the sense of R. R. Coifman and G. Weiss, L a non-negative self-adjoint operator on L^2(𝒳) satisfying the Davies–Gaffney estimate, and X(𝒳) a ball quasi-Banach function space on 𝒳 satisfying some extra mild assumptions. In this article, the authors introduce the Hardy type space H_X, L(𝒳) by the Lusin area function associated with L and establish the atomic and the molecular characterizations of H_X, L(𝒳). As an application of these characterizations of H_X, L(𝒳) , the authors obtain the boundedness of spectral multiplies on H_X, L(𝒳) . Moreover, when L satisfies the Gaussian upper bound estimate, the authors further characterize H_X, L(𝒳) in terms of the Littlewood–Paley functions g_L and g_λ , L^* and establish the boundedness estimate of Schrödinger groups on H_X, L(𝒳) . Specific spaces X(𝒳) to which these results can be applied include Lebesgue spaces, Orlicz spaces, weighted Lebesgue spaces, and variable Lebesgue spaces. This shows that the results obtained in the article have extensive generality.