Sufficient conditions for weighted inequalities of sum form
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.
scientific article; zbMATH DE number 3941833
scientific article