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Hardy spaces associated to self-adjoint operators on general domains

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Abstract

Let \((X,d,\mu )\) be the space of homogeneous type and \(\Omega \) be a measurable subset of X which may not satisfy the doubling condition. Let L denote a nonnegative self-adjoint operator on \(L^2(\Omega )\) which has a Gaussian upper bound on its heat kernel. The aim of this paper is to introduce a Hardy space \(H^1_L(\Omega )\) associated to L on \(\Omega \) which provides an appropriate setting to obtain \(H^1_L(\Omega )\rightarrow L^1(\Omega )\) boundedness for certain singular integrals with rough kernels. This then implies \(L^p\) boundedness for the rough singular integrals, \(1 < p \le 2\), from interpolation between the spaces \(L^2(\Omega )\) and \(H^1_L(\Omega )\). As applications, we show the boundedness for the holomorphic functional calculus and spectral multipliers of the operator L from \(H^1_L(\Omega )\) to \(L^1(\Omega )\) and on \(L^p(\Omega )\) for \(1< p < \infty \). We also study the case of the domains with finite measure and the case of the Gaussian upper bound on the semigroup replaced by the weaker assumption of the Davies–Gaffney estimate.

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Acknowledgements

We would like to thank the referees for careful reading and providing many helpful comments and suggestions, which helps to improve the file.

Funding

Duong and Li are supported by the Australian Research Council through the grant DP 220100285. Lee and Lin are supported by Ministry of Science and Technology, R.O.C. under grant numbers #MOST 110-2115-M-008-009-MY2 and #MOST 109-2115-M-008-002-MY3, respectively. Duong would like to thank the hospitality of National Central University of Taiwan during his visit in 2016 in which this paper started.

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Authors and Affiliations

  1. Department of Mathematics, Macquarie University, Macquarie, NSW, 2109, Australia

    Xuan Thinh Duong & Ji Li

  2. Department of Mathematics, National Central University, Chung-Li, 320, Taiwan, Republic of China

    Ming-Yi Lee & Chin-Cheng Lin

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  1. Xuan Thinh Duong
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Correspondence to Ji Li.

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Duong, X.T., Lee, MY., Li, J. et al. Hardy spaces associated to self-adjoint operators on general domains. Collect. Math. 75, 305–330 (2024). https://doi.org/10.1007/s13348-022-00387-0

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