SCHAUDER ESTIMATES FOR BOUNDARY-LAYER POTENTIALS

Boundary layer potentials with Holder-continuous densities are widely used for representing solutions of boundary value problems for certain partial differential equations. These potentials are of the type upsilon(x)=integral(partial derivative G)(x-xi)rho\x-xi\-\rho\-n+1lambda(xi)domega(xi) on R(n)\partial derivative G, with \rho\ odd and a density lambda of class C(k+alpha). This paper presents a self-contained exposition of the fact that the potential upsilon may be continued as C(k+alpha)-functions onto GBAR and R(n)\GBAR respectively (together with estimates for the Holder-norms) under the least possible regularity assumption for the boundary partial derivative G: it has to be of class C(k+1+alpha).