Fractional integration of summable functions: Maz'ya's~$\Phi$-inequalities

We study the inequalities of the type, where the kernel K is homogeneous of order α -d and possibly vector-valued, the function Φ is positively p-homogeneous, and p = d/(d -α).Under mild regularity assumptions on K and Φ, we find necessary and sufficient conditions on these functions under which the inequality holds true with a uniform constant for all sufficiently regular functions f .