On Seeley-type Universal Extension Operators for the Upper Half Space

TL;DR

This paper introduces a new class of linear extension operators for the upper half space that are bounded across various function spaces, extending the classical reflection method with a more general summation-based approach.

Contribution

It constructs a novel Seeley-type universal extension operator for the upper half space with boundedness in multiple function spaces, including Sobolev, H"older, Besov, and Triebel-Lizorkin spaces.

Findings

The operator is bounded in all $C^k$-spaces.

The operator is bounded in Sobolev and H"older spaces.

The operator is bounded in Besov, Triebel-Lizorkin, and Morrey spaces.

Abstract

Modified from the standard half-space extension via reflection principle, we construct a linear extension operator for the upper half space $\Bbb R^n_+$ that has the form $Ef(x)=\sum_{j=-\infty}^\infty a_jf(x',-b_jx_n)$ for $x_n<0$. We prove that $E$ is bounded in all $C^k$-spaces, Sobolev and H\"older spaces, Besov and Triebel-Lizorkin spaces, along with their Morrey generalizations. We also give an analogous construction on bounded smooth domains.