Heat kernels and Hardy spaces on non-tangentially accessible domains with applications to global regularity of inhomogeneous Dirichlet problems

Abstract

Let n ≥ 2 n\ge 2 and Ω \Omega be a bounded non-tangentially accessible domain (for short, NTA domain) of R n \mathbb {R}^n . Assume that L D L_D is a second-order divergence form elliptic operator having real-valued, bounded, measurable coefficients on L 2 ( Ω ) L^2(\Omega ) with the Dirichlet boundary condition. The main aim of this article is threefold. First, the authors prove that the heat kernels { K t L D } t > 0 \{K_t^{L_D}\}_{t>0} generated by L D L_D are Hölder continuous. Second, for any p ∈ ( 0 , 1 ] p\in (0,1] , the authors introduce the ‘geometrical’ Hardy space H r p ( Ω ) H^p_r(\Omega ) by restricting any element of the Hardy space H p ( R n ) H^p(\mathbb {R}^n) to Ω \Omega and the authors show that, when p ∈ ( n n + δ 0 , 1 ] p\in (\frac {n}{n+\delta _0},1] , H r p ( Ω ) = H p ( Ω ) = H L D p ( Ω ) H^p_r(\Omega )=H^p(\Omega )=H^p_{L_D}(\Omega ) with equivalent quasi-norms, where H p ( Ω ) H^p(\Omega ) and H L D p ( Ω ) H^p_{L_D}(\Omega ) respectively denote the Hardy space on Ω \Omega and the Hardy space associated with L D L_D and where δ 0 ∈ ( 0 , 1 ] \delta _0\in (0,1] is the critical index of the Hölder continuity for the kernels { K t L D } t > 0 \{K_t^{L_D}\}_{t>0} . Third, as applications, the authors obtain the global gradient estimates in both L p ( Ω ) L^p(\Omega ) , with p ∈ ( 1 , p 0 ) p\in (1,p_0) , and H z p ( Ω ) H^p_z(\Omega ) , with p ∈ ( n n + 1 , 1 ] p\in (\frac {n}{n+1},1] , for the inhomogeneous Dirichlet problem of second-order divergence form elliptic equations on bounded NTA domains, where p 0 ∈ ( 2 , ∞ ) p_0\in (2,\infty ) is a constant depending only on n n , Ω \Omega , and the coefficient matrix of L D L_D . Here, the ‘geometrical’ Hardy space H z p ( Ω ) H^p_z(\Omega ) is defined by restricting any element of the Hardy space H p ( R n ) H^p(\mathbb {R}^n) supported in Ω ¯ \overline {\Omega } to Ω \Omega , where Ω ¯ \overline {\Omega } denotes the closure of Ω \Omega in R n \mathbb {R}^n . It is worth pointing out that the range p ∈ ( 1 , p 0 ) p\in (1,p_0) for the global gradient estimate in the scale of Lebesgue spaces L p ( Ω ) L^p(\Omega ) is sharp and the above results are established without any additional assumptions on both the coefficient matrix of L D L_D and the domain Ω \Omega .