A'priori estimates and precise regularity for parabolic systems with discontinuous data

Abstract

We deal with linear parabolic (in sense of Petrovskii) systems of order 2b with discontinuous principal coefficients. A'priori estimates in Sobolev and Sobolev--Morrey spaces are proved for the strong solutions by means of potential analysis and boundedness of certain singular integral operators with kernels of mixed homogeneity. As a byproduct, precise characterization of the Morrey, BMO and H\"older regularity is given for the solutions and their derivatives up to order 2b-1. Some counterexamples of recent publications announcing estimates similar to the obtained here are also given.