# Sharp weighted Sobolev trace inequalities and fractional powers of the Laplacian

## TL;DR

This paper proves sharp weighted Sobolev trace inequalities related to fractional Laplacians, extending previous work by Caffarelli--Silvestre and Yang, with implications for boundary value problems involving fractional operators.

## Contribution

It introduces a new family of sharp Sobolev trace inequalities involving weighted Sobolev norms, connecting fractional Laplacians to Dirichlet-to-Neumann maps in weighted spaces.

## Key findings

- Established sharp weighted Sobolev trace inequalities.
- Linked fractional Laplacians to Dirichlet-to-Neumann operators.
- Generalized previous results by Caffarelli--Silvestre and Yang.

## 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.