Regularity for the CR vector bundle problem II

TL;DR

This paper establishes new regularity estimates for solution operators in the CR setting on strongly pseudoconvex hypersurfaces, improving bounds and applying these results to CR vector bundle integrability and embedding problems.

Contribution

The paper provides explicit Hölder estimates for solution operators of the $ar ext{d}_b$ problem, with improved bounds and regularity conditions, advancing the understanding of CR geometric analysis.

Findings

Derived $ ext{C}^{k+rac{1}{2}}$ Hölder estimates for solution operators

Established $ ext{C}^a$ estimates for solutions with optimal regularity conditions

Applied estimates to improve regularity in CR vector bundle integrability

Abstract

We derive a $\mathcal C^{k+\yt}$ H\"older estimate for $P\phi$, where $P$ is either of the two solution operators in Henkin's local homotopy formula for $\bar\partial_b$ on a strongly pseudoconvex real hypersurface $M$ in $\mathbf C^{n}$, $\phi$ is a $(0,q)$-form of class $\mathcal C^{k}$ on $M$, and $k\geq0$ is an integer. We also derive a $\mathcal C^{a}$ estimate for $P\phi$, when $\phi$ is of class $\mathcal C^{a}$ and $a\geq0$ is a real number. These estimates require that $M$ be of class $\mathcal C^{k+{5/2}}$, or $\mathcal C^{a+2}$, respectively. The explicit bounds for the constants occurring in these estimates also considerably improve previously known such results. These estimates are then applied to the integrability problem for CR vector bundles to gain improved regularity. They also constitute a major ingredient in a forthcoming work of the authors on the local CR embedding problem.