Boundedness of operators with partial integrals with the mixed norm. I

Two types of linear integral operators with partial integrals are considered, which are defined on functions given in a finite rectangle $D=D_1\times D_2$ of the Euclidean point space $\mathbb{R}_2$. Operators of the first type are constructed according to the type of Romanovsky integrals and are studied in the space $C(D_1;L_{p}(D_2))$ norms, space of continuous functions on $\overline{D_1}$ with values in the Lebesgue class $L_p(D_2)$. For general operators, the authors prove that they belong to the class of linear bounded operators from the anisotropic class of functions $L_{p,p^2}$ for $p>1$ to the class of functions with a mixed norm $C (D_1;L_{p}(D_2))$.