- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:47
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
29
2013
1
Hardy spaces and regularity for the inhomogeneous Dirichlet and Neumann problems
Xuan Thinh
Duong
Macquarie University, SYDNEY, NSW, AUSTRALIA
Steve
Hofmann
University of Missouri, COLUMBIA, UNITED STATES
Dorina
Mitrea
University of Missouri, COLUMBIA, UNITED STATES
Marius
Mitrea
University of Missouri, COLUMBIA, UNITED STATES
Lixin
Yan
Zhongshan University, GUANGZHOU, GUANGDONG, CHINA
Hardy space, heat semigroup, atom, inhomogeneous Dirichlet and Neumann problems, Green operator, semiconvex domain, convex domain
This article has three aims. First, we study Hardy spaces, $h^p_L(\Omega)$, associated with an operator $L$ which is either the Dirichlet Laplacian $\Delta_{D}$ or the Neumann Laplacian $\Delta_{N}$ on a bounded Lipschitz domain $\Omega$ in ${\mathbb{R}}^n$, for $0 < p \leq 1$. We obtain equivalent characterizations of these function spaces in terms of maximal functions and atomic decompositions. Second, we establish regularity results for the Green operators, regarded as the inverses of the Dirichlet and Neumann Laplacians, in the context of Hardy spaces associated with these operators on a bounded semiconvex domain $\Omega$ in ${\mathbb{R}}^n$. Third, we study relations between the Hardy spaces associated with operators and the standard Hardy spaces $h^p_r(\Omega)$ and $h^p_z(\Omega)$, then establish regularity of the Green operators for the Dirichlet problem on a bounded semiconvex domain $\Omega$ in ${\mathbb{R}}^n$, and for the Neumann problem on a bounded convex domain $\Omega$ in ${\mathbb{R}}^n$, in the context of the standard Hardy spaces $h^p_r(\Omega)$ and $h^p_z(\Omega)$. This gives a new solution to the conjecture made by D.-C. Chang, S. Krantz and E. M. Stein regarding the regularity of Green operators for the Dirichlet and Neumann problems on $h^p_r(\Omega)$ and $h^p_z(\Omega)$, respectively, for all $\frac{n}{n+1} < p\leq 1$.
Partial differential equations
Fourier analysis
Functional analysis
General
183
236
10.4171/RMI/718
http://www.ems-ph.org/doi/10.4171/RMI/718