A homotopy formula for $a_q$ domains in complex manifolds

We construct a global homotopy formula for $a_q$ domains in a complex manifold. The homotopy operators in the formula will gain $1\mathrm{/}2$ derivative in Hölder-Zygmund spaces ${\mathrm{\Lambda }}^r$ when the boundaries of the domains are in ${\mathrm{\Lambda }}^{r+3}$ with $r>0$.