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\(L^2\) estimates on twisted Cauchy-Riemann complexes. (English) Zbl 1106.32027
Jensen, Gary R. (ed.) et al., 150 years of mathematics at Washington University in St. Louis. Sesquicentennial of mathematics at Washington University, St. Louis, MO, USA, October 3–5, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3603-X/pbk). Contemporary Mathematics 395, 83-103 (2006).
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].

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces