This article is cited in 1 paper

Functional Inequalities and Relative Capacities

V. S. Klimov^a, E. S. Panasenko<sup>b

a</sup> P. G. Demidov Yaroslavl State University
^b Orel State University

Abstract: 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 $.

UDC: 517.518.235

Received: 02.03.1998

DOI: 10.4213/mzm416

 English version: Mathematical Notes, 2002, 72:2, 193–203

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