As a result, we shall formulate problem (\[dbar-neumann\]) as a boundary value problem in which the equation on the domain is of the form $\Box+G_s$ where $\Box$ is the complex Laplacian and $G_s$ is a so-called [*singular Green’s operator*]{} (see [@Grubb]). This result demonstrates a striking difference with the classical case of the $(\dbar,0)$-Neumann problem, where the adjoint is taken with respect to the $L^2$-inner product, and no operator $G_s$ appears. The particular expression of the Hilbert space adjoint $\dbar$, and of the singular Green’s operator $G_s$ arising in the $\dbars$-Neumann problem depend on the choice of the inner product in ${W^s_{(0,q)}(\Omega)}$. We note that with our definition (\[Sobolev\]) we have $$\label{iteration}
\la f,g\ra_s = \la f,g\ra_0 
    + \sum_{j=1}^{2n} \la D_j f,D_j g\ra_{s-1} .$$

Recall that for a $(0,q)$ form $\phi=\sum_{|J|=q}\phi_J d\bar z^J$ with $C^\infty$ coefficients, the operator $\dbar$ is defined as $$\dbar\phi =\sum_{|K|=q+1} \sum_{kJ}\e{K}{kJ} \pd{\phi_J}{\bar z_k} 
d\bar z^K .$$ Then, the [*formal adjoint*]{} $\vartheta$ of $\dbar$ is easily calculated to be $$\vartheta \phi
= -\sum_{|I|=q-1} \e{J}{iI} \pd{\phi_J}{z_i} d\bar z^I.$$

We want to compute the Hilbert space adjoint $\dbar^*$, together with its domain. In carrying out this program, a central role is played by a particular extension of the normal 