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