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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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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.

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Authors and Affiliations

  1. Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Xiaosheng Lin, Dachun Yang & Wen Yuan

  2. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Sibei Yang

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  1. Xiaosheng Lin
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All authors developed and discussed the results and contributed to the final manuscript.

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Correspondence to Dachun Yang.

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This project is partially supported by the National Key Research and Development Program of China (Grant No. 2020YFA0712900), the National Natural Science Foundation of China (Grant Nos. 12371093, 12071197, 12122102 and 12071431), the Fundamental Research Funds for the Central Universities (Grant No. lzujbky-2021-ey18), the Innovative Groups of Basic Research in Gansu Province (Grant No. 22JR5RA391), and the Key Project of Gansu Provincial National Science Foundation (Grant No. 23JRRA1022).

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Lin, X., Yang, D., Yang, S. et al. Hardy Spaces Associated with Non-negative Self-adjoint Operators and Ball Quasi-Banach Function Spaces on Doubling Metric Measure Spaces and Their Applications. Commun. Math. Stat. (2024). https://doi.org/10.1007/s40304-023-00376-0

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