New estimates of Rychkov's universal extension operator for Lipschitz domains and some applications

• Published in: Mathematische Nachrichten 2024-04, Vol.297 (4), p.1407-1443
• Main Authors: Shi, Ziming, Yao, Liding
• Format: Article
• Language: English
• Subjects: Estimates
• ISSN: 0025-584X; 1522-2616
• DOI: 10.1002/mana.202300047

Given a bounded Lipschitz domain Ω⊂Rn$\Omega \subset \mathbb {R}^n$, Rychkov showed that there is a linear extension operator E$\mathcal {E}$ for Ω$\Omega$, which is bounded in Besov and Triebel‐Lizorkin spaces. In this paper, we introduce some new estimates for the extension operator E$\mathcal {E}$ and give some applications. We prove the equivalent norms ∥f∥Apqs(Ω)≈∑|α|≤m∥∂αf∥Apqs−m(Ω)$\Vert f\Vert _{\mathcal A_{pq}^s(\Omega )}\approx \sum _{|\alpha |\le m}\Vert \partial ^\alpha f\Vert _{\mathcal A_{pq}^{s-m}(\Omega )}$ for general Besov and Triebel‐Lizorkin spaces. We also derive some quantitative smoothing estimates of the extended function and all its derivatives on Ω¯c$\overline{\Omega }^c$ up to the boundary.