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

Modified from the standard half-space extension via reflection principle, we construct a linear extension operator for the upper half space ${\mathbb{R}}_{+}^n$ that has the form $Ef(x)={\sum }_{j=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_{j}f(x^{\mathrm{\prime }},-b_j{x}_n)$ for $x_n<0$. We prove that $E$ is bounded in all $C^k$-spaces, Sobolev and Hölder spaces, Besov and Triebel-Lizorkin spaces, along with their Morrey generalizations. We also give an analogous construction on bounded smooth domains.