On Morrey's inequality in Sobolev-Slobodecki\u{\i} spaces

TL;DR

This paper investigates the optimal constant in Morrey's inequality within fractional Sobolev spaces, establishing extremals, analyzing asymptotic behavior, and providing new elementary proofs involving capacity and Hardy's inequality.

Contribution

It generalizes recent work by proving the existence of extremals, analyzing limits of the sharp constant, and introducing a new elementary proof method for Morrey's inequality.

Findings

Existence of extremals for the sharp Morrey constant.

Asymptotic behavior of the constant as s approaches 1 and p approaches infinity.

Convergence of extremals in the limit cases.

Abstract

We study the sharp constant in the Morrey inequality for fractional Sobolev-Slobodecki\u{\i} spaces on the whole $\mathbb{R}^N$. By generalizing a recent work by Hynd and Seuffert, we prove existence of extremals, together with some regularity estimates. We also analyze the sharp asymptotic behaviour of this constant as we reach the borderline case $s\,p=N$, where the inequality fails. This can be done by means of a new elementary proof of the Morrey inequality, which combines: a local fractional Poincar\'e inequality for punctured balls, the definition of capacity of a point and Hardy's inequality for the punctured space. Finally, we compute the limit of the sharp Morrey constant for $s\nearrow 1$, as well as its limit for $p\nearrow \infty$. We obtain convergence of extremals, as well.