Calderón–Zygmund Kernels and Boundary Estimates

The purpose of this chapter is to prove the global a priori estimate (12.3) for the homogeneous Dirichlet problem stated in Theorem 12.2 (see [35, Theorem 4.2]). The desired global a priori estimate (12.3) is a consequence of the explicit boundary representation formula (14.2) for the solutions of the homogeneous Dirichlet problem and an \(L^{p}\) boundedness of some singular integral operators and boundary commutators in the boundary representation formula (14.2) (Theorems 14.2 and 14.5).