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Anisotropic Sobolev inequalities. (English) Zbl 0663.46024

Two main results are proved. Suppose \(p_ j\geq 1\) for \(1\leq j\leq n\), \(\sum_{1\leq j\leq n}1/p_ j>1\) and \(1/q=(\sum 1/p_ j-1)/n\); then there exists a constant C such that the anisotropic Sobolev inequality \(\| u\|_ q\leq C\sum_{1\leq j\leq n}\| D_ ju\|_{p_ j}\) holds for all \(C^{\infty}\) functions u with compact support in \({\mathbb{R}}^ n\) \((\| u\|_ q\) is the \(L^ q({\mathbb{R}}^ n)\) norm of u).
Let \(p_{\alpha}\geq 1\) for each multi-index \(\alpha\) with \(| \alpha | =m\), \(1/q=n^{-m}\sum_{| \alpha | =m}m!/\alpha !p_{\alpha}-m/n\). Suppose that \(\sum_{1\leq j\leq n}1/p_{\beta (j)}>m\) for every \(\beta\) with \(| \beta | =m-1\), \(\beta (j)=(\beta_ 1,...,\beta_{j-1},\beta_ j+1,...,\beta_ n)\); then the following higher order Sobolev inequality holds: \[ \| u\|_ q\leq C\sum_{| \alpha | =m}\| D^{\alpha}u\|_{p_{\alpha}}. \] The proofs proceed by induction and make use of Hölder inequality and permutation inequality for mixed \(L^ p\)-norms [see: A. Benedek and R. Panzone, Duke Math. J. 28, 301-324 (1961; Zbl 0107.089)].
Reviewer: P.Jeanquartier

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D10 Inequalities involving derivatives and differential and integral operators

Citations:

Zbl 0107.089
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