Functional Inequalities and Relative Capacities

In this paper, we study functional inequalities of the form
$$ \|f;Q\| \le C\varphi (\|\nabla f;P\|,\|f;R\|), $$
where $P$, $Q$, and $R$ are Banach ideal spaces of functions on a domain $\Omega \subset \mathbb R^n$, the constant $C$ is the same for all compactly supported functions $f$ satisfying the Lipschitz condition, $\nabla f$ is the gradient of $f$, and $\varphi $ is a continuous degree one homogeneous function. We give compatibility conditions for norms on the spaces $P$, $Q$, and $R$ that ensure the equivalence of the inequality in question to an isoperimetric inequality between the norms of indicators and relative capacities of compact subsets of the domain $\Omega $.