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arXiv · 2110.14477

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

Ziming Shi·Liding Yao
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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 $\|f\|_{\mathscr A_{pq}^s(\Omega)}\approx\sum_{|\alpha|\le m}\|\partial^\alpha f\|_{\mathscr A_{pq}^{s-m}(\Omega)}$ for general Besov and Triebel-Lizorkin spaces. We also derive some quantitative smoothing estimates of the extended function and all its derivatives on $\overline\Omega^c$ up to the boundary.

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BibTeXRIS

Ziming Shi, Liding Yao. 2021-10-27. New Estimates of Rychkov's Universal Extension Operator for Lipschitz Domains and Some Applications. https://doi.org/10.1002/mana.202300047

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