Generalized multiplicative inequalities for ideal spaces

We study the problem of completely describing the domains that enjoy the generalized multiplicative inequalities of the embedding theorem type. We transfer the assertions for the Sobolev spaces $L_p^1(\Omega)$ to the function classes that result from the replacement of $L_p(\Omega)$ with an ideal space of vector-functions. We prove equivalence of the functional and geometric inequalities between the norms of indicators and the capacities of closed subsets of $\Omega$. The most comprehensible results relate to the case of the rearrangement invariant ideal spaces.