Calculus of Variations and Geometric Measure Theory

L. Brasco - F. Prinari - F. Sk

On Morrey's inequality in Sobolev-Slobodeckij spaces

created by brasco on 12 Sep 2023
modified on 14 Sep 2023



Inserted: 12 sep 2023
Last Updated: 14 sep 2023

Pages: 52
Year: 2023


We study the sharp constant in the Morrey inequality for fractional Sobolev-Slobodeckij 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.

Keywords: regularity, fractional Sobolev spaces, Fractional $p-$Laplacian, embeddings, Morrey's inequality, Hardy's inequality, Holder spaces