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

## TL;DR

This paper determines the optimal constant for embedding the Sobolev space $W^{N,1}({
m I ext{R}}^N)$ into $L^ ext{I ext{f}}({
m I ext{R}}^N)$, generalizing previous results to all dimensions using elliptic operator theory.

## Contribution

It extends the known embedding constant results from dimensions one and two to all dimensions by identifying a fundamental solution of a specific elliptic operator.

## Key findings

- Computed the best embedding constant for all dimensions.
- Identified $ ext{log}|x|$ as a fundamental solution of an elliptic operator.
- Extended previous low-dimensional results to general $N$.

## 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 $2N$.