On the Marcinkiewicz–Calderón Interpolation Theorem for Integral Operators

The inverse problem to the classical Marcinkiewicz–Calderón interpolation theorem is considered. Necessary conditions for the Marcinkiewicz–Calderón theorem to hold for the integral operator under consideration are obtained in terms of the kernel of this operator. It is shown that these conditions are sufficient for the given integral operator to be of $(p,q)$-strong type for the same parameters $p$ and $q$ that appear in the interpolation theorem.