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

Abstract

We find necessary and sufficient conditions on the function Φ for the inequalityto be true. Here K is a positively homogeneous of order α − d, possibly vector valued, kernel, Φ is a p-homogeneous function, and p = d/(d − α). The domain Ω ⊂ R d is either bounded with C 1,β smooth boundary for some β > 0 or a halfspace in R d . As a corollary, we describe the positively homogeneous of order d/(d − 1) functions Φ : R d → R that are suitable for the bound Ω Φ(∇u) Ω |∆u|.