MULTILAYER POTENTIALS FOR HIGHER-ORDER SYSTEMS IN ROUGH DOMAINS

We initiate the study of multilayer potential operators associated with any given homogeneous constant-coefficient higher-order elliptic system L in an open set Omega subset of R-n satisfying additional assumptions of a geometric measure theoretic nature. We develop a Calderon-Zygmund-type theory for this brand of singular integral operators acting on Whitney arrays, starting with the case when Omega is merely of locally finite perimeter and then progressively strengthening the hypotheses by ultimately assuming that Omega is a uniformly rectifiable domain (which is the optimal setting where singular integral operators of principal value type are known to be bounded on Lebesgue spaces), and conclude by indicating how this body of results is significant in the context of boundary value problems for the higher-order system L in such a domain Omega.

Automated summary

This study explores the theory of multilayer potential operators associated with any given homogeneous constant-coefficient higher-order elliptic system L in an open set Omega satisfying geometric measure theoretic assumptions. The results indicate that singular integral operators are bounded on Lebesgue spaces in the case of a uniformly rectifiable domain Omega, which is significant for boundary value problems of the higher-order system L in such a domain.