Sobolev and H\"older estimates for homotopy operators of the $\overline\partial$-equation on convex domains of finite multitype

TL;DR

This paper develops optimal Sobolev and H"older estimates for homotopy operators solving the ar-equation on convex domains of finite type, advancing regularity results in complex analysis.

Contribution

It constructs homotopy formulas with optimal regularity estimates for the ar-equation on convex finite type domains, including Sobolev and H"older bounds.

Findings

Homotopy operators achieve fractional Sobolev boundedness $H^{s,p} o H^{s+1/m_q,p}$.

Operators are bounded in H"older-Zygmund spaces $ o \\mathscr C^{s+1/m_q}$.

$L^p$-boundedness results depend on domain type parameters.

Abstract

We construct homotopy formulas for the $\overline\partial$-equation on convex domains of finite type that have optimal Sobolev and H\"older 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 H\"older-Zygmund boundedness $\mathcal H_q:\mathscr C^s\to\mathscr C^{s+1/m_q}$ for all $s\in\mathbb R$ and $1<p<\infty$. We also show the $L^p$-boundedness $\mathcal H_q:H^{s,p}\to H^{s,pr_q/(r_q-p)}$ for all $s\in\mathbb R$ and $1<p<r_q$, where $r_q:=(n-q+1)m_q+2q$.