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Reduced Sobolev inequalities. (English) Zbl 0662.46036

This paper deals with reduced Sobolev inequalities of the type \(\| u\|_ q\leq C\sum_{\alpha \in M}\| D^{\alpha}u\|_ p\) for \(C^{\infty}\) functions u with compact support in \({\mathbb{R}}^ n\). \(\| u\|_ p\) is the \(L^ p({\mathbb{R}}^ n)\) norm of u, \(p\geq 1\). m is an integer satisfying \(1\leq m<n/p\); M is the set of multi-indices \(\alpha\) such that \(| \alpha | =m\) and \(\alpha_ j=0\) or 1, \(1\leq j\leq n\). q is given by \(1/q=1/p-m/n\). The proof proceeds by induction on m and makes use of the technique of mixed \(L^ p\) norms. For some values of m, n, p it is possible to replace M with a suitable proper subset of M.
Reviewer: P.Jeanquartier

MSC:

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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