Dicrete Hölder estimates for a certain kind of parametrix. II

Abstract

In the first paper of this series we have introduced a certain parametrix and the associated potential. The parametrix corresponds to a uniformly elliptic second order differential operator with locally Hölder continuous coefficients in the half-space. Here we show that the potential is an approximate left inverse of the differential operator modulo hyperplane integrals, with the error estimated in terms of the local Hölder norms. As a corollary, we calculate approximately the potential whose density and differential operator originate from the straightening of a special Lipschitz domain. This corollary is aimed for the future derivation of approximate formulae for harmonic functions.