Cauchy integrals, Calderon projectors, and Toeplitz operators on uniformly rectifiable domains

摘要

We develop properties of Cauchy integrals associated to a general class of first-order elliptic systems of differential operators D on a bounded, uniformly rectifiable (UR) domain Omega in a Riemannian manifold M. We show that associated to such Cauchy integrals are analogues of Hardy spaces of functions on Omega annihilated by D, and we produce projections, of Calderon type, onto subspaces of L-P (partial derivative Omega) consisting of boundary values of elements of such Hardy spaces. We consider Toeplitz operators associated to such projections and study their index properties. Of particular interest is a "cobordism argument," which often enables one to identify the index of a Toeplitz operator on a rough UR domain with that of one on a smoothly bounded domain. (C) 2014 Elsevier Inc. All rights reserved.