Regularity estimates for Green operators of Dirichlet and Neumann problems on weighted Hardy spaces
- 23 April 2021
- journal article
- research article
- Published by Mathematical Society of Japan (Project Euclid) in Journal of the Mathematical Society of Japan
- Vol. 73 (2), 597-631
- https://doi.org/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.
Keywords
This publication has 47 references indexed in Scilit:
- Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimatesMemoirs of the American Mathematical Society, 2011
- Riesz transform on manifolds and heat kernel regularityAnnales Scientifiques de lʼÉcole Normale Supérieure, 2004
- Weighted HARDY SpacesMathematische Nachrichten, 1981
- A maximal function characterization of 𝐻^{𝑝} on the space of homogeneous typeTransactions of the American Mathematical Society, 1980
- A decomposition into atoms of distributions on spaces of homogeneous typeAdvances in Mathematics, 1979
- The Atomic Decomposition for Parabolic H p SpacesTransactions of the American Mathematical Society, 1979
- An atomic decomposition of distributions in parabolic Hp spacesAdvances in Mathematics, 1977
- Extensions of Hardy spaces and their use in analysisBulletin of the American Mathematical Society, 1977
- Parabolic maximal functions associated with a distributionAdvances in Mathematics, 1975
- Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. ICommunications on Pure and Applied Mathematics, 1959