Functional inequalities and relative capacities
The authors deal with functional inequalities of the form \[ \|f;Q\|\leq C\varphi\bigl( \|\nabla f;P\|,\|f,R\|\bigr), \tag{1} \] 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 satisfying a Lipschitz condition, \(\nabla f\) is the gradient of a function \(f\), and \(\varphi\) is a continuous degree-one homogeneous function. Compatibility conditions for norms on the spaces \(P,Q\), and \(R\) are given that ensure the equivalence of the functional inequality (1) to a geometric inequality between the norms of indicators and relative capacities of compact subsets of the domain \(\Omega\). The best comprehensible results relate to the case where \(P,Q\) and \(R\) are symmetric spaces.
scientific article; zbMATH DE number 1928277