Sharp weighted Sobolev trace inequalities and fractional powers of the Laplacian

Jeffrey S. Case

doi:10.1016/j.jfa.2020.108567

Sharp weighted Sobolev trace inequalities and fractional powers of the Laplacian

Analysis of PDEs 2022-11-16 v2

Authors: Jeffrey S. Case

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Abstract

We establish a family of sharp Sobolev trace inequalities involving the $W^{k,2}(\mathbb{R}_+^{n+1},y^a)$ -norm. These inequalities are closely related to the realization of fractional powers of the Laplacian on $\mathbb{R}^n=\partial\mathbb{R}_+^{n+1}$ as generalized Dirichlet-to-Neumann operators associated to powers of the weighted Laplacian in upper half space, generalizing observations of Caffarelli--Silvestre and of Yang.

Keywords

sobolev trace inequalities fractional laplacian weighted norm dirichlet-to-neumann operator upper half space sharp inequality

Cite

@article{arxiv.1901.09843,
  title  = {Sharp weighted Sobolev trace inequalities and fractional powers of the Laplacian},
  author = {Jeffrey S. Case},
  journal= {arXiv preprint arXiv:1901.09843},
  year   = {2022}
}

Comments

Corrected proof of Lemma 2.4; 26 pages

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