On the Cauchy Problem for the Dolbeault Complex in the Sobolev spaces

Let $D$ be a bounded domain in $\mathbb C^n$ ($n>1$) with a twice smooth boundary $\partial D$. We describe necessary and sufficient Cauchy problem's solvability conditions for the Dolbeault complex in the space of differential forms of bidegree $(0,q)$, $0<q<n$, with coefficients from the Sobolev space $H^1(D)$ in the domain $D$.