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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Weighted local Orlicz-Hardy spaces on domains and their applications in inhomogeneous Dirichlet and Neumann problems


Authors: Jun Cao, Der-Chen Chang, Dachun Yang and Sibei Yang
Journal: Trans. Amer. Math. Soc. 365 (2013), 4729-4809
MSC (2010): Primary 42B35; Secondary 42B30, 42B20, 42B25, 35J25, 42B37, 47B38, 46E30
Published electronically: February 27, 2013
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Abstract: Let $ \Omega $ be either $ \mathbb{R}^n$ or a strongly Lipschitz domain of $ \mathbb{R}^n$, and $ \omega \in A_{\infty }(\mathbb{R}^n)$ (the class of Muckenhoupt weights). Let $ L$ be a second-order divergence form elliptic operator on $ L^2 (\Omega )$ with the Dirichlet or Neumann boundary condition, and assume that the heat semigroup generated by $ L$ has the Gaussian property $ (G_1)$ with the regularity of their kernels measured by $ \mu \in (0,1]$. Let $ \Phi $ be a continuous, strictly increasing, subadditive, positive and concave function on $ (0,\infty )$ of critical lower type index $ p_{\Phi }^-\in (0,1]$. In this paper, the authors first introduce the ``geometrical'' weighted local Orlicz-Hardy spaces $ h^{\Phi }_{\omega ,\,r}(\Omega )$ and $ h^{\Phi }_{\omega ,\,z}(\Omega )$ via the weighted local Orlicz-Hardy spaces $ h^{\Phi }_{\omega }(\mathbb{R}^n)$, and obtain their two equivalent characterizations in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by $ L$ when $ p_{\Phi }^-\in (n/(n+\mu ),1]$. Second, the authors furthermore establish three equivalent characterizations of $ h^{\Phi }_{\omega ,\,r}(\Omega )$ in terms of the grand maximal function, the radial maximal function and the atomic decomposition when the complement of $ \Omega $ is unbounded and $ p_{\Phi }^-\in (0,1]$. Third, as applications, the authors prove that the operators $ \nabla ^2{\mathbb{G}}_D$ are bounded from $ h^{\Phi }_{\omega ,\,r}(\Omega )$ to the weighted Orlicz space $ L^{\Phi }_{\omega }(\Omega )$, and from $ h^{\Phi }_{\omega ,\,r}(\Omega )$ to itself when $ \Omega $ is a bounded semiconvex domain in $ \mathbb{R}^n$ and $ p_{\Phi }^-\in (\frac {n}{n+1},1]$, and the operators $ \nabla ^2{\mathbb{G}}_N$ are bounded from $ h^{\Phi }_{\omega ,\,z}(\Omega )$ to $ L^{\Phi }_{\omega }(\Omega )$, and from $ h^{\Phi }_{\omega ,\,z}(\Omega )$ to $ h^{\Phi }_{\omega ,\,r}(\Omega )$ when $ \Omega $ is a bounded convex domain in $ \mathbb{R}^n$ and $ p_{\Phi }^-\in (\frac {n}{n+1},1]$, where $ {\mathbb{G}}_D$ and $ {\mathbb{G}}_N$ denote, respectively, the Dirichlet Green operator and the Neumann Green operator.


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Additional Information

Jun Cao
Affiliation: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
Email: caojun1860@mail.bnu.edu.cn

Der-Chen Chang
Affiliation: Department of Mathematics and Department of Computer Science, Georgetown University, Washington, DC 20057
Email: chang@georgetown.edu

Dachun Yang
Affiliation: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
Email: dcyang@bnu.edu.cn

Sibei Yang
Affiliation: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
Email: yangsibei@mail.bnu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05832-1
PII: S 0002-9947(2013)05832-1
Keywords: Strongly Lipschitz domain, semiconvex domain, convex domain, divergence form elliptic operator, Dirichlet Green operator, Neumann Green operator, Dirichlet boundary condition, Neumann boundary condition, weight, local Orlicz-Hardy space, Gaussian property, nontangential maximal function, Lusin area function, atom.
Received by editor(s): July 25, 2011
Published electronically: February 27, 2013
Additional Notes: The second author was partially supported by an NSF grant DMS-1203845 and a Hong Kong RGC competitive earmarked research grant $#$601410.
The third (corresponding) author was supported by the National Natural Science Foundation of China (Grant No. 11171027) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20120003110003).
Article copyright: © Copyright 2013 American Mathematical Society