## 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:

 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

### Keywords:

reduced Sobolev inequalities; mixed $$L^ p$$ norms
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