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


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