This excellent survey article concerns the so-called twisted Cauchy-Riemann complex, with some applications. This twisted complex can be defined as follows: Let \(\tau \geq 0\) be a nonnegative function on a domain \(\Omega\) in \(\mathbb{C}^n\). Consider then \(\Lambda^{0,0}(\Omega) \mathop{\to}\limits^{T} \Lambda^{0, 1}(\Omega) \mathop{\to}\limits^{S} \Lambda^{0,2}(\Omega)\) where \(Tf = \bar\partial (\sqrt\tau f) \; f\in \Lambda^{0,0} (\Omega) = \{u \in \Lambda^{0, 1}(\bar\Omega) \langle u, \bar\partial r \rangle = 0\; \text{on}\; b \Omega\}\) and \(S u = \sqrt \tau \bar \partial (u)\), \(u \in \Lambda^{0, 1} (\Omega)\) where as usual \(\Lambda^{p, q}(\Omega)\) denotes the \((p, q)\) forms with \(\mathcal{C}^\infty(\Omega)\) components. The complex’s natural continuation would involve pre- and post-multiplying by \(\displaystyle\frac{1}{\sqrt\tau}\) of the two next form level. Suppose now that \(\Omega\) is a smoothly bounded, pseudoconvex domain, \(A\) is an arbitrary possitive function \(\Omega\). Then for any form \(u \in \mathcal{D}^{0, 1}\) one has the basic estimate \[ \| \sqrt\tau \bar\partial u\| ^2_\lambda + \| (\sqrt{A + \tau}) \bar\partial_\lambda^* u \| ^2_\lambda\geq \]
\[ \int_\Omega \{ \tau i \partial\bar\partial \lambda (u,u) - i \partial\bar\partial(u,u) - \displaystyle\frac{i \partial \tau \Lambda \bar \partial c(u, u)}{A}\} e^{-\lambda} \] where \((u, v)_\lambda = \displaystyle\sum_{k = 1}^n \int_\Omega u_k \cdot \bar v_k e^{-\lambda}\), and \(\bar\partial_\lambda^*\) is the adjoint with respect with this inner product to \(\bar\partial\). If one denotes the expression within the brackets in the right-hand side of the basic estimate by \(\Theta (u, u) (= \Theta_{\lambda, \tau, A}(u,u))\), there is the possibility of mixing plurisubharmonicity on \(\rho\lambda\) with plurisuperharmonicity condition on \(\tau\) to achieve \(\Theta > 0\). By duality arguments one has the following result: Suppose \(\Omega\) pseudoconvex, \(\Theta > 0\), and \(\varphi\) an arbitrary plurisubharmonic function (eventually \(\varphi = 0\)). Then, for any \(\alpha \in L^2_{(0, 1)} (\Omega)\) such that \(\bar\partial \alpha = 0\) there exists a solution to \(\bar\partial u = \alpha\) satisfying the estimate \(\int_\Omega | u| ^2 \displaystyle\frac{e^{-\lambda}}{f + \tau} e^{-\varphi} \leq C \int_\Omega [\Theta]^{-1} (\alpha, \alpha) e^{- \lambda} e^{-\varphi}\) with an arbitrary constant \(C\). The author gives some interesting applications, using the basic estimate. In the case \(\lambda\) is plurisubharmonic, \(\tau = e^{-\lambda}\) and \(A = 2 \tau\), \(\Theta\) is positive if \(\lambda\) has selfbounded gradient (i.e., \(i \partial \lambda \bar\partial \Lambda \bar\partial \lambda (u, u) \leq i \partial \bar\partial \lambda (u, u)\; u \in \mathbb{C}^n\)). In this case the above existence results gives a solution satisfying \(\int_\Omega | u| ^2 e^{-\varphi}\leq 6 \int_\Omega [i \partial \bar\lambda]^{-1} (\alpha, \alpha) e^{-\varphi}\). The applications given in the paper concerns invariant metric estimates, local estimates for the \(\bar\partial\)-Neumann problem, global estimate for the \(\bar\partial\)-Neumann problem, extension results. The author carefully explains the notations behind the computations, and the difficulties which appear.
For the entire collection see [Zbl 1086.01003].