## A sharp inequality of J. Moser for higher order derivatives.(English)Zbl 0672.31008

Let m, n be two integers with $$n\geq 2$$ and $$1\leq m<n$$. Let us set $$p=n/m$$ and denote by q the conjugate exponent. For a function $$u\in C^ m({\mathbb{R}}^ n)$$ with support in a domain $$\Omega$$ of finite Lebesgue measure say $$| \Omega |$$, we denote by $$\| \nabla^ mu\|_ p$$ the $$L_ p$$ norm of the m-th order gradient of u. Then the author proves the following Theorem:
There exists constants c(m,n) and $$\beta_ 0(m,n)$$ such that for any $$u\in C^ m({\mathbb{R}}^ n)$$ with $$\| \nabla^ mu\|_ p\leq 1$$ one has: $\int_{\Omega}\exp (\beta | u(x)|^ q)dx\leq c(m,n)| \Omega |$ for all $$\beta \leq \beta_ 0(m,n)$$. If $$\beta >\beta_ 0(m,n)$$ then one can find u for which the above integral can be made as large as desired.
Moreover the exact value of $$\beta_ 0(m,n)$$ is explicitely computed. This theorem extends a result of J. Moser proved in the particular case $$m=1$$. The key is to write the function u as a Riesz potential and first prove the theorem in this situation. Then the general case follows.
Reviewer: J.Lacroix

### MSC:

 31B15 Potentials and capacities, extremal length and related notions in higher dimensions 31C25 Dirichlet forms
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