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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

MSC:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Keywords:
weight functions; Sobolev inequality
Full Text:
References:
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