Maz'ya's $\Phi$-inequalities on domains

We find necessary and sufficient conditions on the function $\Phi$ for the inequality $$\Big|\int_\Omega \Phi(K*f)\Big|\lesssim \|f\|_{L_1(\mathbb{R}^d)}^p$$ to be true. Here $K$ is a positively homogeneous of order $\alpha - d$, possibly vector valued, kernel, $\Phi$ is a $p$-homogeneous function, and $p=d/(d-\alpha)$. The domain $\Omega\subset \mathbb{R}^d$ 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 $d/(d-1)$ functions $\Phi\colon \mathbb{R}^d \to \mathbb{R}$ that are suitable for the bound $$\Big|\int_\Omega \Phi(\nabla u)\Big|\lesssim \int_\Omega |\Delta u|.$$