Regularity for Robin boundary problems of Laplace equations and Hardy spaces on C1 and (semi-)convex domains

Let n >= 3 and St be a bounded C-1 or (semi-)convex domain in R-n. In this article, the authors establish W-1,W- P-estimates for inhomogeneous Robin problems of Laplace equations in Omega for any p is an element of (1, infinity). As an application, the alpha-order Holder regularity of the heat kernels, generated by the Laplace operator Delta(R) on Omega with the Robin boundary condition, is obtained for any alpha is an element of (0, 1). Moreover, the authors prove that the Riesz transform del Delta(-1/2)(R) is bounded on L-P(Omega) for any p is an element of (1, infinity), and from the Hardy space H-Delta R(p) (Omega) to the Lebesgue space L-P (Omega) for any p is an element of (0, 1], or to the "geometric" Hardy space H-r(P) (Omega) for any p is an element of (n/n+ 1 , 1], where H-Delta R(p) (Omega) denotes the Hardy space associated with the operator Delta R. Furthermore, the inclusion relations H-Delta N(p) (Omega) subset of H-Delta R(p) (Omega) subset of H-Delta R(p) (Omega) for any p is an element of (n/n+1, 1] are obtained, where H-Delta N(p) (Omega) and H-Delta D(p) (Omega) denote the Hardy spaces associated, respectively, with the Laplace operators Delta(N) and Delta(D) on Omega with the Neumann and the Dirichlet boundary conditions. Meanwhile, when Omega is (semi-)convex, the boundedness of the second-order Riesz transform del(2)Delta(-1)(R) from H-Delta R(p) (Omega) to L-P (Omega) for any p is an element of (0, 1], or to the Hardy space H-r(P) (Omega) for any p is an element of (n/n+1, 1], is also established. As applications, W-2,W- P type estimates for the inhomogeneous Robin problem of Laplace equations with any p is an element of (0, 1] are given. When St is (semi-)convex, the obtained results in this article for the L-P-boundedness of the Riesz transform essentially improve the known results via extending the range p is an element of (1, 3 + epsilon) into p is an element of (1, infinity), where epsilon is an element of (0, infinity) is a positive constant depending on Omega (C) 2021 Elsevier Inc. All rights reserved.