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

Let \(({\mathcal {X}},d,\mu )\) be a doubling metric measure space in the sense of R. R. Coifman and G. Weiss, L a non-negative self-adjoint operator on \(L^2({\mathcal {X}})\) satisfying the Davies–Gaffney estimate, and \(X({\mathcal {X}})\) a ball quasi-Banach function space on \({\mathcal {X}}\) satisfying some extra mild assumptions. In this article, the authors introduce the Hardy type space \(H_{X,\,L}({\mathcal {X}})\) by the Lusin area function associated with L and establish the atomic and the molecular characterizations of \(H_{X,\,L}({\mathcal {X}}).\) As an application of these characterizations of \(H_{X,\,L}({\mathcal {X}})\), the authors obtain the boundedness of spectral multiplies on \(H_{X,\,L}({\mathcal {X}})\). Moreover, when L satisfies the Gaussian upper bound estimate, the authors further characterize \(H_{X,\,L}({\mathcal {X}})\) in terms of the Littlewood–Paley functions \(g_L\) and \(g_{\lambda ,\,L}^*\) and establish the boundedness estimate of Schrödinger groups on \(H_{X,\,L}({\mathcal {X}})\). Specific spaces \(X({\mathcal {X}})\) 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.