CONDITION R AND FEFFERMAN ’ S MAPPING THEOREM

It is shown that Fefferman's mapping theorem extends to the case of manifolds, that is a biholomorphic map between two strictly 
pseudoconvex manifolds extends smoothly to their boundaries.


Introduction.
A central question in complex analysis is "does every proper holomorphic mapping f : D → D of bounded domains D, D with smooth boundaries in C n extend smoothly to the boundary of D?" The answer has been known to be "yes" in dimension one for a long time.In higher dimensions, in case D and D are strictly pseudoconvex and f is biholomorphic, Fefferman's famous mapping theorem [8] answers the question in the affirmative.
Bell and Ligocka [4] simplified the proof of Fefferman's mapping theorem and extended the theorem to a wide class of pseudoconvex domains.
In [3], Fefferman's mapping theorem was extended to smoothly bounded pseudoconvex subdomains of Stein manifolds that satisfy condition R. Thereafter, the question was asked whether all smoothly bounded pseudoconvex domains satisfy condition R. Recently, Barrett [2] and Christ [5] have shown that this question has an answer in the negative.But that was not the end of condition R, because the case of strictly pseudoconvex manifolds that are not Stein had not been determined.At first it was thought (because of the work of Barrett [1]) that one could not do without the assumption of Steinness.
In this note, we show that a strictly pseudoconvex manifold need not be Stein before it satisfies condition R; and following the work of Bedford et al. [3], we extend Fefferman's mapping theorem to all strictly pseudoconvex manifolds.

Preliminaries.
Let Ω be a relatively compact domain in an n-dimensional complex manifold X.The space L 2 (n,0) (Ω) is defined to be the set of (n, 0) forms ω such that is finite.The space L 2 (n,0) (Ω) is a Hilbert space with inner product given by 2) The Bergman-Kobayashi projection P Ω associated to Ω is the orthogonal projection of L 2 (n,0) (Ω) onto H (n,0) (Ω), the closed subspace of L 2 (n,0) (Ω) consisting of holomorphic (n, 0) forms.If Ω has a smooth boundary, Ω satisfies condition R if the Bergman-Kobayashi projection associated to Ω maps C ∞ (n,0) ( Ω) into C ∞ (n,0) ( Ω).To make use of the proof in [3], we show that if Ω above has smooth boundary and it is strictly pseudoconvex, then Ω satisfies condition R; and, in addition, if p 0 is a point in X near the boundary ∂Ω of Ω, then there are n functions g 1 ,...,g n that are holomorphic in a neighborhood of Ω and that form a coordinate system at p 0 .
Our main result is the following theorem.

Condition R.
To establish condition R for smoothly bounded strictly pseudoconvex subdomains of complex manifolds, we need a result of Gunning and Rossi [9] which we met on the way to proving theorems in [6,7].Their result is the following theorem.
Theorem 3.1.Let Ω be a strictly pseudoconvex domain in a complex manifold Y .There are a Stein manifold X and a proper holomorphic mapping π : Ω → X with the following properties: (i) π : ᏻ X ᏻ Ω ; (ii) there are finitely many points x 1 ,...,x z in X such that π −1 (x j ) is a compact subvariety of Ω of positive dimension, and π : Ω \ π −1 (x j ) X \{x 1 ,...,x z }.The first statement means that the rings of holomorphic functions ᏻ X and ᏻ Ω on X and Ω, respectively, are isomorphic under the map induced by π .The second means that Ω \∪π −1 (x j ) and X \{x 1 ,...,x z } are biholomorphic.Now from the proof of Theorem 3.1 as given in [9], it is clear that there is a strictly pseudoconvex neighborhood Ω of Ω such that Ω can replace Ω in Theorem 3.1 so that the compact set ∪π −1 (x j ) corresponding to Ω is contained in Ω.
If X corresponds to Ω in Theorem 3.1 and X = π(Ω), then clearly if Ω has a smooth boundary then X is a Stein strictly pseudoconvex manifold with a smooth boundary, and therefore, as is well-known, X satisfies condition R.
We can regard Ω as imbedded in X.Then it is clear that Therefore the Bergman-Kobayashi projections P X and P Ω are equal, and it is not difficult to see (using Sobolev spaces) that Ω satisfies condition R. Let Ω be a strictly pseudoconvex subdomain of a complex manifold Y .Then near the boundary ∂Ω of Ω, local coordinates are given by holomorphic functions in a neighborhood of Ω.
Proof.As indicated in Section 3, from the proof of Theorem 3.1 as given in [9] it is clear that there is a strictly pseudoconvex neighborhood Ω of Ω such that Ω can replace Ω in Theorem 3.1 so that the compact set ∪π −1 (x j ) corresponding to Ω is contained in Ω.Now if p 0 is a point in Ω near the boundary ∂Ω of Ω, let π(p 0 ) have holomorphic functions g 1 ,...,g n on the Stein manifold X that form local coordinates at π(p 0 ).Then g 1 • π,...,g n • π form local coordinates at p 0 , which are holomorphic in a neighborhood of Ω.

Proof of Theorem 2.1.
The proof of Theorem 2.1 relies on the following two lemmas whose proofs are in [3].
that vanishes to at most finite order at any boundary point of Ω 2 , then f * ω vanishes to at most finite order at any boundary point of Ω 1 .Now to prove Theorem 2.1, we initiate the proof of Theorem 2.1 in [3]: Let p 0 be a boundary point of Ω 1 and let z 1 ,...,z n be holomorphic coordinates near p 0 .We show that f extends smoothly to ∂Ω 1 near p 0 .Let {p i } be a sequence of points in Ω 1 that converges to p 0 .Then {f (p i )} converges to a point q 0 in ∂Ω 2 .Let g 1 ,...,g n be n functions on Ω 2 that extend to be holomorphic in a neighborhood of Ω2 in X 2 and that form a coordinate chart at q 0 .Define a holomorphic function u near p 0 via (5.1)By Lemmas 5.1 and 5.2 u extends smoothly to ∂Ω 1 near p 0 and u vanishes to a finite order near p 0 .If α = (α 1 ,...,α n ) is a multi-index, then we define g α = n i=1 g α i i .Lemma 5.1 implies that the form f * (g α dg 1 ∧ ••• ∧ dg n ) extends smoothly to ∂Ω 1 near p 0 for each α.Hence, u and u(g α •f ) extend smoothly to ∂Ω 1 near p 0 for each α, and u vanishes to at most finite order at p 0 .By the division theorem cited in [3], g i •f extends smoothly to ∂Ω 1 near p 0 for each i.Hence f extends smoothly to ∂Ω 1 near p 0 .Since p 0 was arbitrarily chosen, we conclude that f extends smoothly to all of ∂Ω 1 .Now we can replace f by f −1 and then the theorem follows.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

4 .
Local coordinates near the boundary.Again from Theorem 3.1 we get the last theorem that we need in the proof of Theorem 2.1.Theorem 4.1.

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation