JournalsrmiVol. 29, No. 1pp. 183–236

Hardy spaces and regularity for the inhomogeneous Dirichlet and Neumann problems

  • Xuan Thinh Duong

    Macquarie University, Sydney, Australia
  • Steve Hofmann

    University of Missouri, Columbia, USA
  • Dorina Mitrea

    University of Missouri, Columbia, United States
  • Marius Mitrea

    University of Missouri, Columbia, USA
  • Lixin Yan

    Zhongshan University, Guangzhou, China
Hardy spaces and regularity   for  the inhomogeneous Dirichlet and Neumann problems cover
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Abstract

This article has three aims. First, we study Hardy spaces, hLp(Ω)h^p_L(\Omega), associated with an operator LL which is either the Dirichlet Laplacian ΔD\Delta_{D} or the Neumann Laplacian ΔN\Delta_{N} on a bounded Lipschitz domain Ω\Omega in Rn{\mathbb{R}}^n, for 0<p10 < 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 Ω\Omega in Rn{\mathbb{R}}^n. Third, we study relations between the Hardy spaces associated with operators and the standard Hardy spaces hrp(Ω)h^p_r(\Omega) and hzp(Ω)h^p_z(\Omega), then establish regularity of the Green operators for the Dirichlet problem on a bounded semiconvex domain Ω\Omega in Rn{\mathbb{R}}^n, and for the Neumann problem on a bounded convex domain Ω\Omega in Rn{\mathbb{R}}^n, in the context of the standard Hardy spaces hrp(Ω)h^p_r(\Omega) and hzp(Ω)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 hrp(Ω)h^p_r(\Omega) and hzp(Ω)h^p_z(\Omega), respectively, for all nn+1<p1\frac{n}{n+1} < p\leq 1.

Cite this article

Xuan Thinh Duong, Steve Hofmann, Dorina Mitrea, Marius Mitrea, Lixin Yan, Hardy spaces and regularity for the inhomogeneous Dirichlet and Neumann problems. Rev. Mat. Iberoam. 29 (2013), no. 1, pp. 183–236

DOI 10.4171/RMI/718