  We study regularity of solutions $u$ to $\overline\partial u=f$ on a
relatively compact $C^2$ domain $D$ in a complex manifold of dimension $n$,
where $f$ is a $(0,q)$ form. Assume that there are either $(q+1)$ negative or
$(n-q)$ positive Levi eigenvalues at each point of boundary $\partial D$. Under
the necessary condition that a locally $L^2$ solution exists on the domain, we
show the existence of the solutions on the closure of the domain that gain
$1/2$ derivative when $q=1$ and $f$ is in the H\"older-Zygmund space
$\Lambda^r(\overline D)$ with $r>1$. For $q>1$, the same regularity for the
solutions is achieved when $\partial D$ is either sufficiently smooth or of
$(n-q)$ positive Levi eigenvalues everywhere on $\partial D$.
