Best constants for two families of higher order critical Sobolev embeddings

TL;DR

This paper determines the best constants in higher order Sobolev inequalities, including those embedding into $L^ abla$ and exponential integrability, advancing understanding of critical Sobolev embeddings.

Contribution

It provides explicit optimal constants for higher order Sobolev inequalities in critical cases, extending previous results and including new embeddings into larger target spaces.

Findings

Optimal constants for embeddings into $L^ abla$ spaces.

Existence of sharp constants in exponential Sobolev inequalities.

Extension of known inequalities to higher order derivatives.

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