## 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

Zbl 0107.089
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