Regularity of convolution type operators with PC symbols in Bessel potential spaces over two finite intervals

Abstract

AbstractConvolution type operators acting between Bessel potential spaces defined on a union of two finite intervals are studied from the point of view of their regularity properties. The operators are assumed to have kernels with Fourier transforms in the class of piecewise continuous matrix functions which is a frequent situation in several problems from mathematical physics. Various Wiener‐Hopf operators having the same regularity properties as the considered convolution type operators are presented. Moreover, besides a criterion for the Fredholm property, sufficient conditions for the invertibility of the operators in study are obtained. The approach is based on the consideration of relations between different kinds of operators. In this way, the transfer of regularity properties, between the related operators, appears as a natural consequence. An application to a wave diffraction problem with impedance conditions is illustrated. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)