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

Abstract

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.

Keywords

Hardy space / Ball quasi-Banach function space / Non-negative self-adjoint operator / Atom / Molecule / Schrölder group / Spectral multiplier / Littlewood–Paley function

Funding

National Key Research and Development Program of China(2020YFA0712900)

National Natural Science Foundation of China(11971058, 12071197, 12122102 and 12071431)

Fundamental Research Funds for the Central Universities(lzujbky-2021-ey18)

Innovative Groups of Basic Research in Gansu Province(22JR5RA391)

Key Project of Gansu Provincial National Science Foundation(23JRRA1022)