Maz'ya's Φ-inequalities on domains

We find necessary and sufficient conditions on the function Φ for the inequality |∫_Ω Φ(K*f)| \|f\|L₁(Rᵈ)ᵖ to 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⊂ Rᵈ is either bounded with C1,β smooth boundary for some β > 0 or a halfspace in Rᵈ. As a corollary, we describe the positively homogeneous of order $d/(d-1)$ functions Φ Rᵈ → R that are suitable for the bound |∫_Ω Φ(∇ u)| ∫_Ω |Δ u|.