Functional inequalities and generalized capacities

In this criteria were found for the validity of a functional inequality of the form $\|f;Q\| \leqslant C\|\nabla f;P\|$, where $P$ and $Q$ are normed ideal spaces of functions on a domain $\Omega \subset \mathbb R^n$, and the constant $C$ is the same for compactly supported functions $f$ satisfying a Lipschitz condition. Conditions for norm agreement in the space $P$ and $Q$ are given under which the functional inequality in question is equivalent to a geometric inequality relating the $Q$-norms of the indicators and $P$-capacities of compact subset of $\Omega$. Estimates are given and general properties of the capacities are studied.