On a class of integral equations of convolution type and its applications

Operators of convolution type with variable coefficients are investigated in spaces of series $E$ commonly employed for such operators and supplemented by the spaces $M_0^{\sup}$, $M_0^{\mathrm{mes}}$, $M^{\sup}$, $M^{\mathrm{mes}}$ of bounded measurable functions. The properties of these new spaces are investigated. Conditions insuring that operators are of the Noether type are investigated and the index of operators with coefficients from these spaces are calculated. A theorem asserting the coincidence of zeros in these spaces is stated. The results are applied to investigate conditions for integral equations with homogeneous kernels to be of Noether type. For the last class the structure of zeros is clarified in the case of constant coefficients.