Preprint

Inserted: 19 apr 2018
Last Updated: 19 apr 2018

Pages: 16
Year: 2018

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 \[ \vert\vert u \vert\vert_{L^\infty} \leq c_k \int \vert \nabla^k (-\Delta)^{(N-k)/2} u\vert \] for all $u \in C^\infty_c(\mathbb{R}^N)$ (the case $k=N$ was handled in previous work of the first author). 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 \[ \int_{\Omega} \exp (\beta_0 \vert u \vert^{p^\prime}) \leq A \vert \Omega \vert \] for all $u$ such that $\vert\vert \nabla^m u\vert\vert_{L^p(\Omega)} \leq 1$, where $\vert\vert \nabla^m u\vert\vert_{L^p(\Omega)}$ is the traditional semi-norm on the space $W^{m,p}(\Omega)$.

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