Calderón–Zygmund Kernels and Interior Estimates

ThisInterior estimate chapter is devoted to the proof of Theorem 12.1 (Theorem 13.3) that is based on some local interior a priori estimates for the solutions of theinterior a priori estimate homogeneous Dirichlet problem (Lemma 13.2). The main idea of proof may be considered as an integral perturbation about the constant coefficient case, which goes back to Eugenio Elia Levi [89] (Theorem 13.1). The vanishing mean oscillation (VMO) assumption on the coefficients is of the greatest relevance in the study of an error term expressed by singular commutators (Corollary 11.7). The desired interior a priori estimate (12.2) follows in a standard way from Lemma 13.2. by a covering argument.