arXiv subjects
Publications and source records attributed to S. M. Webster.
We give a new solution to the local integrability problem for CR vector bundles over strictly pseudoconvex real hypersurfaces of dimension seven or greater. It is based on a KAM rapid convergence argument and avoids the previous more difficult Nash-Moser methods. The solution is sharp as to Hölder continuity.
We derive a $\mathcal C^{k+\yt}$ Hölder estimate for $Pϕ$, 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 $\mathbf C^{n}$, $ϕ$ 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 $Pϕ$, when $ϕ$ 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.
We consider a formally integrable, strictly pseudoconvex CR manifold $M$ of hypersurface type, of dimension $2n-1\geq7$. Local CR, i.e. holomorphic, embeddings of $M$ are known to exist from the works of Kuranishi and Akahori. We address the problem of regularity of the embedding in standard Hölder spaces $C^{a}(M)$, $a\in\mathbf{R}$. If the structure of $M$ is of class $C^{m}$, $m\in\mathbf{Z}$, $4\leq m\leq\infty$, we construct a local CR embedding near each point of $M$. This embedding is of class $C^{a}$, for every $a$, $0\leq a < m+(1/2)$. Our method is based on Henkin's local homotopy formula for the embedded case, some very precise estimates for the solution operators in it, and a substantial modification of a previous Nash-Moser argument due to the second author.