Maz'ya's $\mathrm{\Phi }$-inequalities on domains

We find necessary and sufficient conditions on the function $\mathrm{\Phi }$ for the inequality $$\mid {\int }_{\mathrm{\Omega }}\mathrm{\Phi }(K\ast f)\mid \lesssim \mathrm{\parallel }f{\mathrm{\parallel }}_{L_1({\mathbb{R}}^d)}^p$$ to be true. Here $K$ is a positively homogeneous of order $\alpha -d$, possibly vector valued, kernel, $\mathrm{\Phi }$ is a $p$-homogeneous function, and $p=d\mathrm{/}(d-\alpha )$. The domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$ is either bounded with $C^{1,\beta }$ smooth boundary for some $\beta >0$ or a halfspace in ${\mathbb{R}}^d$. As a corollary, we describe the positively homogeneous of order $d\mathrm{/}(d-1)$ functions $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:{\mathbb{R}}^d\to \mathbb{R}$ that are suitable for the bound $$\mid {\int }_{\mathrm{\Omega }}\mathrm{\Phi }(\mathrm{\nabla }u)\mid \lesssim {\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\Delta }u\mathrm{\mid }\mathrm{.}$$