The following new sufficient condition for exact regularity is given: Let \(\Omega \) be a smooth bounded pseudoconvex domain in \(\mathbb{C}^n,\) \(\rho \) a defining function for \(\Omega .\) Let \(1\leq q \leq n.\) Assume that there is a constant \(C\) such that for all \(\epsilon >0\) there exist a defining function \(\rho_\epsilon \) for \(\Omega \) and a constant \(C_\epsilon \) with \(1/C < | \nabla \rho_\epsilon | < C \) on \(b\Omega,\) and \[ \left \| \sum_{| K| =q-1} \, ' \left ( \sum_{j,k=1}^n \frac{\partial^2 \rho_\epsilon}{\partial z_j \partial \overline{z_k}}\, \frac{\partial \rho}{\partial \overline{z_j}}\, \overline{u_{kK}} \right ) d\overline{z_K} \right \| ^2 \leq \epsilon (\| \overline \partial u \| ^2 + \| \overline \partial^*u\| ^2) + C_\epsilon \| u\| _{-1}^2 \] for all \(u\in \mathcal{C}^\infty _{(0,q)} (\overline \Omega ) \cap {\text{dom}} (\overline \partial^* ).\) Then the \(\overline \partial\)- Neumann operator \(N_q\) on \((0,q)\) forms is exactly regular in Sobolev norms, that is \(\| N_q u\| _s \leq C_s \| u\| _s,\) for \(s\geq 0\) and all \(u\in W_{(0,q)}^s (\Omega ).\)
In this way a theory of global regularity of the \(\overline \partial\)- Neumann operator is developed which unifies the two principal approaches to date, namely the one via compactness due to Kohn-Nirenberg and Catlin and the one via plurisubharmonic defining functions and/or vector fields that commute approximately with \(\overline \partial\) due to Boas and Straube. In the final section, the author gives some interesting estimates of operators arising from the regularized \(\overline \partial\)- Neumann problem.