  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.

Complex Variables