Best constants for two families of higher order critical Sobolev embeddings

Abstract: In this paper we obtain the best constants in some higher order Sobolev inequalities in the critical exponent. These inequalities can be separated into two types: those that embed into $L^\infty(\mathbb{R}^N)$ and those that embed into slightly larger target spaces. Concerning the former, we show that for $k \in {1,\ldots, N-1}$, $N-k$ even, one has an optimal constant $c_k>0$ such that [ |u|{L^\infty} \leq c_k \int |\nabla^k (-\Delta)^{(N-k)/2} u|] for all $u \in C^\infty_c(\mathbb{R}^N)$ (the case $k=N$ was handled in a paper by Shafrir). Meanwhile the most significant of the latter is a variation of D. Adams' higher order inequality of J. Moser: For $\Omega \subset \mathbb{R}^N$, $m \in \mathbb{N}$ and $p=\frac{N}{m}$, there exists $A>0$ and optimal constant $\beta_0>0$ such that [ \int{\Omega} \exp (\beta_0 |u|^{p^\prime}) \leq A |\Omega| ] for all $u$ such that $|\nabla^m u|{L^p(\Omega)} \leq 1$, where $|\nabla^m u|{L^p(\Omega)}$ is the traditional semi-norm on the space $W^{m,p}(\Omega)$.