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 $BV$.