The best constant in the embedding of $W^{N,1}({\mathbb{R}}^N)$ into $L^{\infty }({\mathbb{R}}^N)$

Abstract

We compute the best constant in the embedding of $W^{N,1}({\mathbb{R}}^N)$ into $L^{\infty }({\mathbb{R}}^N)$, extending a result of Humbert and Nazaret in dimensions one and two to any $N$. The main tool is the identification of $\log |x|$ as a fundamental solution of a certain elliptic operator of order $2N$.