On the boundedness of the integral convolution operator in a pair of classical Lebesgue spaces $L_p$ and $L_r$

In terms of the kernel of an integral convolution operator, a constructive criterion for its boundedness in a pair of classical Lebesgue spaces $L_p$ and $L_r$ is obtained. It is shown that in order for the integral convolution operator to act boundedly from $L_p$ to $L_{r,p}$, it is necessary and sufficient that the kernel $K(t)$ of the operator belonged to the Marcinkiewicz space $M_{t^{1-1/q}}$.