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

Abstract

We construct homotopy formulas for the $\overline{\partial }$-equation on convex domains of finite type that have optimal Sobolev and Hölder estimates. For a bounded smooth finite type convex domain $\Omega \subset {\mathbb{C}}^n$ that has $q$-type $m_q$ for $1\le q\le n$, our $\overline{\partial }$ solution operator ${\mathcal{H}}_q$ on $(0,q)$-forms has (fractional) Sobolev boundedness ${\mathcal{H}}_q:H^{s,p}\to H^{s+1/m_q,p}$ and Hölder-Zygmund boundedness ${\mathcal{H}}_q:{\mathcal{C}}^s\to {\mathcal{C}}^{s+1/m_q}$ for all $s\in \mathbb{R}$ and $1<p<\infty$. We also show the $L^p$-boundedness ${\mathcal{H}}_q:H^{s,p}\to H^{s,p{r}_q/(r_q-p)}$ for all $s\in \mathbb{R}$ and $1<p<r_q$, where $r_q:=(n-q+1)m_q+2q$.

35 pages. Add Appendix B which is not contained in the journal version. The Sobolev estimate part of Theorem B.1 can be found in the preprint arXiv:2502.00925

References in corpus (3)

• New Estimates of Rychkov's Universal Extension Operator for Lipschitz Domains and Some Applications
• Some Intrinsic Characterizations of Besov-Triebel-Lizorkin-Morrey-type Spaces on Lipschitz Domains
• On Seeley-type Universal Extension Operators for the Upper Half Space