Accepted Paper
Inserted: 12 sep 2023
Last Updated: 26 jul 2024
Journal: J. Funct. Anal.
Pages: 52
Year: 2023
Abstract:
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
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