A sharp higher order Sobolev embedding

TL;DR

This paper establishes precise Sobolev embedding results from higher order Sobolev spaces into L^1 spaces, identifying extremal functions and improving previous constant estimates.

Contribution

It provides sharp embedding theorems for Sobolev spaces into L^1 and determines the extremal functions, refining earlier bounds by Kalyabin.

Findings

Sharp embedding constants for W^{k,2}_0(-1,1) into L^1(-1,1)

Identification of extremal functions for these embeddings

Improved bounds over previous estimates by Kalyabin

Abstract

We obtain sharp embeddings from the Sobolev space $W^{k,2}_0(-1,1)$ into the space $L^1(-1,1)$ and determine the extremal functions. This improves on a previous estimate of the sharp constants of these embeddings due to Kalyabin.