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

TL;DR

This paper investigates inequalities involving fractional integrals of summable functions, establishing necessary and sufficient conditions for their validity under mild regularity assumptions on kernels and functions.

Contribution

It provides a comprehensive characterization of Maz'ya's $\

Findings

Identifies conditions for fractional integral inequalities to hold.

Establishes a link between kernel regularity and inequality validity.

Provides a framework for analyzing vector-valued kernels.

Abstract

We study the inequalities of the type $|\int_{\mathbb{R}^d} \Phi(K*f)| \lesssim \|f\|_{L_1(\mathbb{R}^d)}^p$, 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 $f$.