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Integral representations of differential forms on Cauchy-Riemann manifolds and the theory of CR-functions. (English. Russian original) Zbl 0589.32035

Russ. Math. Surv. 39, No. 3, 41-118 (1984); translation from Usp. Mat. Nauk 39, No. 3(237), 39-106 (1984).
The work gives a description of the current state of the theory of CR forms (in particular CR functions) on CR manifolds. Such manifold \(M\) is given locally by \[ M=\{z\in {\mathbb C}^ n,\;\rho_ 1(z)=\ldots=\rho_ k(z)=0\}, \] where \(\rho_ s\) are smooth, real functions in a domain \(\Omega \subset {\mathbb C}^ n\). In the case \(M\) is a hypersurface or completely real submanifold of i.e. \(k=1\) or \(k=n\), the theory of CR forms is well understood, and the authors recall shortly well known results proved by Andreotti, Hill, Kohn, Naruki, Rossi and others.
Therefore the authors consider the general case, mainly of so called \(q\)-concave CR manifolds. Recall briefly this notion. Let \(T^ c_{\tau}(M)\) be the largest complex linear subspace in the real tangent space \(T_{\tau}(M)\) at \(\tau\in M\). The Levi form of \(M\) at \(\tau\in M\) is given by the quadratic form \(L_{\tau}(M)\) on \(T^ c_{\tau}(M)\) with values in the normal space \(N_{\tau}\) at \(\tau\) : \[ L_{\tau}(M)(\xi)=- \sum^{k}_{s=1}\left(\sum_{\alpha,\beta}\frac{\partial^ 2\rho}{\partial z_{\alpha}\quad \partial \bar z_{\beta}}(\xi)\xi_{\alpha}{\bar \xi}_{\beta}\right)\cdot \text{grad}\, \rho_ s. \] A CR manifold \(M\) is called \(q\)-concave, if for every \(\tau\in M\) and any \(x\in\mathbb R^ k\setminus \{0\}\) the scalar form \(\langle x_{\tau},L_{\tau}(M)(\xi)\rangle\), \(x_{\tau}=\sum^{k}_{s=1}x_ s \text{grad}\, \rho_ s(\tau),\) \(\xi \in T^ c_{\tau}(M)\), has at least \(q\) negative eigenvalues.
The importance of this notion is clear by the following result proved in the work: If \(\Omega\) is pseudoconvex and \(M\) is a \(q\)-concave manifold, then there exists a neighborhood \(\Omega'\subset \Omega\) of \(M\) such that for any \(r\), \(0\leq r<q\) or \(n-k\geq r>n-k-q\) and any CR form \(f\in C^{(s)}_{0,r}(M)\), \(s\geq 0\), with coefficients in \(C^{(s)}(M)\), there exists a \({\bar \partial}\)-closed form \(F\in C_{0,r}^{(s'- )}(\Omega ')\), \(s'<s\), such that \(F\wedge {\bar \partial}\rho_ 1\wedge...\wedge {\bar \partial}\rho_ k\in C^{(s)}(\Omega ')\) and \(f=F\) on \(\Omega'\cap M\).
The main tool in proving this result and many other is given by various integral formulas (representations) for CR forms. They are too complicated to be described here. In the paper they are cleverly applied in several places. For example, to solve locally tangential Cauchy-Riemann equations on \(q\)-concave manifolds (Theorem 4.2.1). Moreover, the same method of integral representation is successfully applied to CR functions on 1-concave manifolds (Theorem 5.2.1). The work also contains more precise results on integral representations of CR functions on standard CR manifolds. Namely for \[ M=\{z,\quad z=(u,v)\in {\mathbb C}^ k\times {\mathbb C}^{n-k},\quad \rho =\text{Im}\, u-F(v,v)=0\}, \] where \(F=(F_ 1,...,F_ k)\) is a \({\mathbb C}^ k\)-valued Hermitian form on \({\mathbb C}^{n-k}\) (Theorem 5.3.1).
The power of integral representation technique for differential forms becomes clear by the authors’ consideration of non-smooth differential forms. This enables them to prove in the second part of this paper [Mat. Sb., Nov. Ser. 127(169), No. 1(5), 92–112 (1985; Zbl 0589.32036)] \(L^{\infty}\) and \(L^ 1\) type estimates for the solution of Cauchy-Riemann equation \({\bar \partial}_ Mf=g\); here \(g\) is a CR form with bounded or integrable coefficients.
The work should be familiar to everybody working in the theory of CR forms and CR manifolds.
Reviewer: J. Janas

MSC:

32V40 Real submanifolds in complex manifolds
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32D15 Continuation of analytic objects in several complex variables
32F10 \(q\)-convexity, \(q\)-concavity

Citations:

Zbl 0589.32036
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