### Regularity for the CR vector bundle problem IIRegularity for the CR vector bundle problem II

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 Author Gong, Xianghong ♦ Webster, S. M. Source arXiv.org Content type Text File Format PDF Date of Submission 2009-11-24 Language English
 Subject Domain (in DDC) Natural sciences & mathematics ♦ Mathematics Subject Keyword Mathematics - Complex Variables ♦ 32V05 ♦ math 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. Educational Use Research Learning Resource Type Article