A condition for the compactness of operators in a certain class and its application to the analysis of the solubility of non-local problems for elliptic equations

A class of “integral” operators arising in the analysis of non-local problems in which the values of a solution at the boundary of the domain under consideration are expressed through its values at interior points is investigated. These operators are defined in terms of measures close to Carleson measures. A condition ensuring the complete continuity of such operators is found. This result enables one to complement and extend results on the Fredholm property of a broad class of non-local problems for a second-order elliptic equation.