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Superposition of imbeddings and Fefferman’s inequality. (English) Zbl 0948.46023

The aim of this paper is to establish simple conditions for the weight function \(v\), guaranteeing validity of the inequality \[ \Big(\int_Bu^2(x) v(x) dx\Big)^{1/2}\leq c\Big(\int_B|\nabla u(x)|^2 dx\Big)^{1/2},\tag{1} \] where \(B\) is a ball in \(\mathbb R^n\) and \(c\) is a positive constant independent of \(u\in W_0^{1,2}(B)\). As a method of finding such conditions the authors used an idea of a decomposition of the embedding which is represented by (1) into an embedding of \(W_0^{1,2}(B)\) to a suitable target space \(X\) and an embedding from \(X\) to \(L^2_v(B)\).
Some applications of the result are presented.
Reviewer: P.Gurka (Praha)

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D15 Inequalities for sums, series and integrals
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