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Sufficient conditions for weighted inequalities of sum form. (English) Zbl 0587.26011
The authors discuss inequalities of the form (for y in a class $${\mathcal D}$$ of functions with domain I): (1) There is a constant K such that for all $$y\in {\mathcal D},$$ $\int_{I}N| y^{(i)}|^ p\leq K\{\int_{I}W| y|^ p+\int_{I}P| y^{(n)}|^ p\}.$ (2) For each $$\epsilon >0$$ there is a number K($$\epsilon)$$ such that for all $$y\in {\mathcal D}$$, $\int_{I}N| y^{(i)}|^ p\leq K(\epsilon)\int_{I}W| y|^ p+\epsilon \int_{I}P| y^{(n)}|^ p.$ (3) There exist $$\xi\geq 0$$ $$(\xi >0$$ for $$j\neq 0)$$, $$\eta >0$$, $$K>0$$, and a set $$\Gamma$$ of positive numbers such that for all $$\epsilon \in \Gamma$$ and $$y\in {\mathcal D},$$ $\int_{I}N| y^{(j)}|^ p\leq K\{\epsilon^{-\xi}\int_{I}W| y|^ p+\epsilon^{\eta}\int_{I}P| y^{(n)}|^ p\}.$ In these inequalities I is an interval of the real line, $$1\leq p<\infty,$$ $$0\leq j<n,$$ and N,W,P are positive measurable functions (which satisfy some additional conditions). For $$N=P=W=1,$$ the $$p=\infty$$ case of (3) is also considered.
Reviewer: J.Pečarić

##### MSC:
 26D10 Inequalities involving derivatives and differential and integral operators
##### Keywords:
weighted integral inequalities; sum form
Full Text:
##### References:
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