# ON THE OPERATORS WITH PARTIAL INTEGRALS IN THE FUNCTION SPACES OF TWO VARIABLES.

Linear operators with partial integrals are studied. Using Banach’s closed graph

theorem, a general theorem on the continuity acting from a space $X$ to a space $Y$ of linear operator

$K$ with partial integrals is proved. Here $X$ and $Y$ are complete metric spaces of measurable

functions with a shift-invariant metric, and the space $X$ contains, together with each function,

its modulus. With the application of this theorem, the continuity acting of the operator $K$ in

various function spaces is established. The conditions of this theorem are not satisﬁed by spaces

of continuously diﬀerentiable functions. In this connection, a theorem on continuity acting of the

operator $K$ in spaces of continuously diﬀerentiable functions is established. The conditions for

continuity acting of the operator $K$ from the spaces of continuously diﬀerentiable functions to

various classes of function spaces are obtained. The continuity of the operator $K$ deﬁned on the

space $BV$ of bounded variation functions of two variables is proved, and the acting conditions

for this operator in the space $BV$ of functions deﬁned on a ﬁnite rectangle are established.

**Keywords**: linear operators with partial integrals, Banach’s closed graph theorem, acting and

continuity of the operators, function spaces, the space $BV$ of bounded variation functions,

conditions for the action in $BV$.