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April, 2021 Regularity estimates for Green operators of Dirichlet and Neumann problems on weighted Hardy spaces
The Anh BUI, Xuan Thinh DUONG
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J. Math. Soc. Japan 73(2): 597-631 (April, 2021). DOI: 10.2969/jmsj/83938393

Abstract

In this paper we first study the generalized weighted Hardy spaces $H^{p}_{L,w}(X)$ for $0 < p \le 1$ associated to nonnegative self-adjoint operators $L$ satisfying Gaussian upper bounds on the space of homogeneous type $X$ in both cases of finite and infinite measure. We show that the weighted Hardy spaces defined via maximal functions and atomic decompositions coincide. Then we prove weighted regularity estimates for the Green operators of the inhomogeneous Dirichlet and Neumann problems in suitable bounded or unbounded domains including bounded semiconvex domains, convex regions above a Lipschitz graph and upper half-spaces. Our estimates are in terms of weighted $L^{p}$ spaces for the range $1 < p <\infty$ and in terms of the new weighted Hardy spaces for the range $0 < p \le 1$. Our regularity estimates for the Green operators under the weak smoothness assumptions on the boundaries of the domains are new, especially the estimates on Hardy spaces for the full range $0 < p \le 1$ and the case of unbounded domains.

Funding Statement

The second author was supported by the Australian Research Council through the research grant ARC DP190100970.

Citation

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The Anh BUI. Xuan Thinh DUONG. "Regularity estimates for Green operators of Dirichlet and Neumann problems on weighted Hardy spaces." J. Math. Soc. Japan 73 (2) 597 - 631, April, 2021. https://doi.org/10.2969/jmsj/83938393

Information

Received: 19 December 2019; Published: April, 2021
First available in Project Euclid: 18 March 2021

Digital Object Identifier: 10.2969/jmsj/83938393

Subjects:
Primary: 35J25
Secondary: 35J08, 42B30, 42B35, 42B37

Rights: Copyright ©2021 Mathematical Society of Japan

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Vol.73 • No. 2 • April, 2021
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