Inversion of Some Potential-Type Operators with Oscillating Kernels in the Elliptic and Non-Elliptic Cases

Abstract

Within the framework of the method of approximative inverse operators we construct the inversion of potential-type operators in R n with kernels aðjtjÞe ijtj =jtj nÀa , 0 < Re a < n both in the elliptic and non-elliptic cases. The characteristic aðrÞ, being sufficiently smooth in some neighbourhood of the point r ¼ 1, locally belongs to L 1 . The non-elliptic case, in which the symbol of the corresponding potential simultaneously has zeroes and singularities, ''spread'' over some spheres in R n , is the most difficult.