  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 $f$.
