  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 H\"older
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.
