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Modified from the standard half-space extension via reflection principle, weconstruct 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. Wealso give an analogous construction on bounded smooth domains.

On Seeley-type Universal Extension Operators for Upper Half Space