Heat Kernels and Hardy Spaces on Non-Tangentially Accessible Domains with Applications to Global Regularity of Inhomogeneous Dirichlet Problems

Abstract

Let n ≥ 2 and Ω be a bounded non-tangentially accessible domain (for short, NTA domain) of R n . Assume that L D is a second-order divergence form elliptic operator having realvalued, bounded, measurable coefficients on L 2 (Ω) with the Dirichlet boundary condition. The main aim of this article is threefold. First, the authors prove that the heat kernels {K L D t } t>0 generated by L D are Hölder continuous. Second, for any p ∈ (0, 1], the authors introduce the 'geometrical' Hardy space H p r (Ω) by restricting any element of the Hardy space H p (R n ) to Ω, and show that, when p ∈with equivalent quasi-norms, where H p (Ω) and H p L D (Ω) respectively denote the Hardy space on Ω and the Hardy space associated with L D , and δ 0 ∈ (0, 1] is the critical index of the Hölder continuity for the kernels {K L D t } t>0 . Third, as applications, the authors obtain the global gradient estimates in both L p (Ω), with p ∈ (1, p 0 ), and H p z (Ω), with p ∈ ( n n+1 , 1], for the inhomogeneous Dirichlet problem of second-order divergence form elliptic equations on bounded NTA domains, where p 0 ∈ (2, ∞) is a constant depending only on n, Ω, and the coefficient matrix of L D . Here, the 'geometrical' Hardy space H p z (Ω) is defined by restricting any element of the Hardy space H p (R n ) supported in Ω to Ω, where Ω denotes the closure of Ω in R n . It is worth pointing out that the range p ∈ (1, p 0 ) for the global gradient estimate in the scale of Lebesgue spaces L p (Ω) is sharp and the above results are established without any additional assumptions on both the coefficient matrix of L D , and the domain Ω.