is a multi-index and we are using standard multi-index notation. Moreover, $\CP:=|\alpha|!/\alpha!$ denotes the polynomial coefficient. \[The naturality of this choice of the Sobolev inner product will be pointed out and discussed below.\]

We define the Sobolev space $W^s (\Omega)$ to be the closure of $C^\infty (\bar \Omega)
$ with respect to the above inner product. We denote by ${W^s_{(0,q)}(\Omega)}$ the space of $(0,q)$ forms whose coefficients are in ${W^s(\Omega)}$. If $\phi=\sum_{|J|=q}\phi_J d\bar z^J$ and $\psi=\sum_{|J|=q}\psi_Jd\bar z^J$, then the inner product in ${W^s_{(0,q)}(\Omega)}$ is defined by $$\la \phi,\psi\ra_s := \sum_{|J|=q}\sum_
{|\alpha|\le s} \CP
\int_\Omega D^\alpha \phi_J \overline{D^\alpha \psi_J} ,$$ where we use the standard notation $J$ to denote a $q$-vector with increasing entries, and $\alpha$ to denote a multi-index. \[Note that the inner product of forms of different degrees is defined to be 0.\]

For a $(0,q)$ form $\phi=\sum_{|J|=q}\phi_J d\bar z^J$ with $C^\infty$ coefficients, the operator $\dbar$ is defined by $$\label{d-bar}
\dbar\phi :=\sum_{|K|=q+1} \sum_{kJ}\e{K}{kJ} \pd{\phi_J}{\bar z_k}  
d\bar z^K ,$$ where $\e{K}{kJ}$ equals the sign of the permutation $kJ\mapsto K$ if $\{ k\}\cup J=K$ as sets, and is $0$ otherwise. We continue to use $\overline{\partial}$ to denote its closure in the $W^s$ topology. In this way, for each integer $q=0,1,\dots,n$, we 