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Sobolev inequalities for products of powers. (English) Zbl 0686.46020
The paper gives sufficient conditions on a pair of weight functions u, v in \({\mathbb{R}}^ n\), \(n>1\), so that the Sobolev inequality \[ (\int | f(x)|^ q u(x)dx)^{1/q}\leq C(\int | \nabla f(x)|^ p v(x)dx)^{1/p} \] holds for every \(f\in C^{\infty}_ 0({\mathbb{R}}^ n)\) with \(1<p\leq q<\infty\). Given a ball B in \({\mathbb{R}}^ n\), 2B denotes the ball concentric with B, whose radius is twice that of B. It is supposed that \(u(2B)\leq cu(B)\) for all \(B\subset {\mathbb{R}}^ n\), with \(u(B)=\int_{B}u(x)dx.\) For example the Sobolev inequality holds if \(p<q\), \(| B|^{1/n} u(B)^{1/q}\leq cv(B)^{1/p}\)
and \(v(x)=w(x)(1+| x|)^ s g_ 1(x)...g_ m(x)\) where w is a weight function such that \(w(B)(\int_{B}w(x)^{-1/(p-1)}dx)^{p-1}\leq c| B|^ p,\) \(s\geq 0\) and \(g_ j(x)=| x-a_ j|^{b(j)}(1+| x-a_ j|)^{-b(j)},\) b(j)\(\geq 0\), \(a_ 1,...,a_ m\) distinct points in \({\mathbb{R}}^ n\).
Reviewer: P.Jeanquartier

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text: DOI
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