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

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发表日期：

2024年1月19日

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Xiaosheng Lin,
Dachun Yang,
Sibei Yang,
Wen Yuan

摘要

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