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

Abstract

We study the inequalities of the type $|{\int }_{{\mathbb{R}}^d}\Phi (K\ast f)|\lesssim \Vert f{\Vert }_{L_1({\mathbb{R}}^d)}^p$, where the kernel $K$ is homogeneous of order $\alpha -d$ and possibly vector-valued, the function $\Phi$ is positively $p$-homogeneous, and $p=d/(d-\alpha )$. Under mild regularity assumptions on $K$ and $\Phi$, we find necessary and sufficient conditions on these functions under which the inequality holds true with a uniform constant for all sufficiently regular functions $f$.

21 pages, 2 figures