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Estimates for the \(\bar \partial\)-Neumann problem in pseudoconvex domains of finite type in \(\mathbb{C}^ 2\). (English) Zbl 0821.32011

This paper gives detailed proofs for the results which were announced by the authors in Proc. Natl. Acad. Sci. USA 85, No. 23, 8771-8774 (1988; Zbl 0662.32015).
The authors construct a parametrix for the \(\overline \partial\)-Neumann problem for bounded pseudoconvex domains \(\Omega\) of finite type in \(\mathbb{C}^ 2\). They use this to obtain sharp regularity results for the Neumann operator \(N\) and for solutions of \(\overline \partial u = f\), with \(f\) a (0,1)-form on \(\Omega\).
The authors take the approach of P. C. Greiner and E. M. Stein [‘Estimates for the \(\overline \partial\)-Neumann problem’. Princeton Univ. Press. Princeton, N.J. (1977; Zbl 0354.35002)], who solved analogous problems for strongly pseudoconvex domains in complex manifolds. In particular, an intrinsic pseudodifferential operator formula for the Dirichlet to Neumann operator is obtained and is used to construct the boundary operator \(\square^ +\). Another boundary operator \(\square^ -\) is obtained as the corresponding operator for the domain \(\mathbb{C}^ 2 \setminus \Omega\). All of the above constructions are valid for any smooth bounded domain in \(\mathbb{C}^ 2\). The hypothesis of finite type is used to study the invertibility of \(\square^ +\) and of \(\square^ -\) and to establish that \(\square^ - \square^ + \approx \square_ b\). Solving the \(\overline \partial\)-Neumann problem on \(\Omega\) is reduced to inverting the boundary operator \(\square_ b\). An explicit formula for a parametrix for the \(\overline \partial\)-Neumann problem is obtained and the commutation properties of the components of the parametrix are used to establish regularity properties for \(N\) and for solutions of \(\overline \partial u = f\).
The paper concludes with the proof of an extension of the Henkin-Skoda theorem to bounded domains of finite in \(\mathbb{C}^ 2\). The theorem states that the zero variety \(Z\) of a holomorphic function is the zero variety of a function in the Nevanlinna class if and only if \(Z\) satisfies the Blaschke condition.

MSC:

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
35S05 Pseudodifferential operators as generalizations of partial differential operators
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