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On weighted Sobolev interpolation inequalities. (English) Zbl 0815.26007

Denote by \(\Lambda^ k(\mathbb{R}^ n)\) the collection of all functions \(f\) on \(\mathbb{R}^ n\) which have weak derivatives \(D^ \alpha f\), \(|\alpha|\leq k\). Let \(1\leq p\leq q< \infty\), \(v\in A_ p\), and let \(w\) be a doubling weight such that \[ \| f- f_ Q\|_{L^ q_ w(Q)}\leq C_ 0 w(Q)^{1/q} v'(Q)^{1/p'} \ell(Q)^{1- n} \|\nabla f\|_{L^ p_ v(Q)} \] holds for all cubes \(Q\) in \(\mathbb{R}^ n\) and all \(f\in \Lambda^ 1(\mathbb{R}^ n)\). Here, \(p'= p/(p- 1)\), \(v'= v^{1/(1- p)}\), \(f_ Q= | Q|^{- 1} \int_ Q f dx\), and \(\ell(Q)= | Q|^{1/n}\). Let \(i,k\in \mathbb{N}\), \(1\leq i< k\), and \(0< a< 1\). Then \[ \| \nabla^ i f\|_{L^ q_ w(\mathbb{R}^ n)}\leq C\| f\|^{1- a}_{L^ p_ v(\mathbb{R}^ n)} \cdot\|\nabla^ k f\|^ a_{L^ p_ v(\mathbb{R}^ n)} \] for all \(f\in \Lambda^ k(\mathbb{R}^ n)\) with \(\| \nabla^ k f\|_{L^ p_ v(\mathbb{R}^ n)}\neq 0\) if and only if \(\ell(Q)^{ka/i} w(Q)^{1/q}\leq Cv(Q)^{1/p}\) for all cubes \(Q\) in \(\mathbb{R}^ n\). This is one of the results proved in the paper. The others are its modifications.
Reviewer: B.Opic (Praha)

MSC:

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