New Estimates of Rychkov's Universal Extension Operator for Lipschitz Domains and Some Applications

Given a bounded Lipschitz domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, Rychkov showed that there is a linear extension operator $\mathcal{E}$ for $\mathrm{\Omega }$ which is bounded in Besov and Triebel-Lizorkin spaces. In this paper we introduce some new estimates for the extension operator $\mathcal{E}$ and give some applications. We prove the equivalent norms $|f|{\mathscr A{pq}^s(\Omega)}\approx\sum_{|\alpha|\le m}|\partial^\alpha f|{\mathscr A{pq}^{s-m}(\Omega)}$ for general Besov and Triebel-Lizorkin spaces. We also derive some quantitative smoothing estimates of the extended function and all its derivatives on ${\bar{\mathrm{\Omega }}}^c$ up to the boundary.