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

Abstract

Given a bounded Lipschitz domain $\Omega \subset {\mathbb{R}}^n$, Rychkov showed that there is a linear extension operator $\mathcal{E}$ for $\Omega$ 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 $\Vert f{\Vert }_{{\mathcal{A}}_{pq}^s(\Omega )}\approx {\sum }_{|\alpha |\le m}\Vert {\partial }^{\alpha }f{\Vert }_{{\mathcal{A}}_{pq}^{s-m}(\Omega )}$ for general Besov and Triebel-Lizorkin spaces. We also derive some quantitative smoothing estimates of the extended function and all its derivatives on ${\overline{\Omega }}^c$ up to the boundary.

Final version; 33 pages. Add a footnote for Remark 6.9 and edit the acknowledgement. Appeared in Mathematische Nachrichten

References in corpus (3)

• Nazarov's uncertainty principles in higher dimension
• Characterisation by Local Means of Anisotropic Lizorkin--Triebel Spaces with Mixed Norms
• Some Intrinsic Characterizations of Besov-Triebel-Lizorkin-Morrey-type Spaces on Lipschitz Domains

Cited by in corpus (1)

• Sobolev and Hölder estimates for homotopy operators of the $\overline{\partial }$-equation on convex domains of finite multitype