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 recent 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 $W^{m,p}(\Omega)$.