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Sufficient conditions for three weight sum inequalities in Lebesgue spaces. (English) Zbl 0879.26053

Denote by \(|u|_{p,w}\) the “usual” norm of a function \(u\) in the weighted Lebesgue space \(L^p(I,w);\) here \(I=(a,b)\) with \(-\infty \leq a<b\leq \infty\), \(w\) is a weight on \(I\) and \(p\in (1,\infty)\). Using an elementary approach based on a convenient form of unweighted interpolation inequality, the authors derive sufficient conditions for the validity of the inequality \[ |u^{(j)}|_{q,w} \leq C\big (|u|_{r,w_0} +|u^{(m)}|_{p,w_m}\big) \] on a certain class of sufficiently smooth functions \(u\) (with a constant \(C\) independent of \(u\)) for various choices of parameters \(p,q,r\in (1,\infty)\) and \(0\leq j<m\). These conditions on weights can be easily verified and they are “not far” from necessary ones, which follows from given examples.
Reviewer: B.Opic (Praha)

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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