On generalized Romanovsky operators with partial integrals in the space of continuous functions

The paper contains sufficient conditions for the action of generalized and linear generalized partial integral Romanovsky operator in the space of continuous functions defined on an n-dimensional parallelepiped. Continuity of these operators is established in case of their action in the space of continuous functions and, in a more general case of continuous kernels of operators with values in the space of measurable and Lebesgue integrable functions. Estimates are obtained for the norms of the operators mentioned. The dependence of the estimate for the norm of a linear Romanovsky type operator with generalized partial integrals on the space dimension and on the norm of continuous kernels of generalized partial integral Romanovsky operators with values in the space of measurable and Lebesgue integrable functions is shown. The properties established are applied to the study of linear generalized partial integral equations of Romanovsky type, in particular, to the study of generalized partial integral equation of n-connected Markov chains.