On the analyticity of CR mappings.

*(English)*Zbl 0583.32021The paper deals with the real-analyticity of a CR diffeomorphism J from a real-analytic CR manifold M onto another one M’. Results of this type were found previously by H. Lewy [Atti Accad. Naz. Lincei, Rend., Cl. Sci. Fis. Mat. Nat., VIII. Ser. 35, 1-8 (1967)] and S. I. Pincuk [Math. USSR, Sb. 27(1975), 375-392 (1977); translation from Mat. Sb., n. Ser. 98(140), 416-435 (1975; Zbl 0366.32010)] with M and M’ being strictly pseudoconvex and J is of class \(C^ 1\). A \(C^{\infty}\) version of this result is found by L. Nirenberg, S. Webster and P. Yang in Commun. Pure Appl. Math. 33, 305-338 (1980; Zbl 0436.32018) and a related result, valid in codimension higher than one, is due to S. M. Webster [Complex analysis of several variables, Proc. Symp., Madison/Wis. 1982, Proc. Symp. Pure Math. 41, 199-208 (1984; Zbl 0568.32013)].

The present paper treats the general case in which M and M’ are generic real-analytic submanifolds of \({\mathbb{C}}^{n+d}\) of codimension \(d\geq 1\). The method of proof is based on a version of the edge of the wedge theorem where the standard complex structure is replaced by a new one, which allows the embedding of M in \({\mathbb{C}}^{n+d}\) as \({\mathbb{C}}^ n\times {\mathbb{R}}^ d\), but whose underlying real-analytic structure is the same as the standard one. This procedure resembles one used previously by M. Derridj [Invent. Math. 79, 197-215 (1985; Zbl 0538.32009)] and it enables the authors to use arguments of holomorphic extension from \({\mathbb{R}}^ d\) to \({\mathbb{C}}^ d\). A key in this procedure is an identity that involves the tangential Cauchy-Riemann vector fields and the defining equations of the manifolds. This identity is used to apply the above edge of the wedge theorem and to show that the components of the diffeomorphism under study are analytic. Using these facts the authors are able to show that J is real-analytic in the following cases: when M and M’ are hypersurfaces in \({\mathbb{C}}^{n+1}\) and M’ is essentially finite; when M and M’ are hypersurfaces of finite type in \(C^ 2\); when M and M’ are contained in compact real-analytic subvarieties of \({\mathbb{C}}^{n+1}\). The last assertion is based in part on a result of K. Diederich and J. Fornaess [Ann. Math., II. Ser. 107, 371-384 (1978; Zbl 0378.32014)] and has some consequences about global analyticity. For example, if M and M’ are the real-analytic boundaries of two bounded open subsets \(\Omega\) and \(\Omega\) ’ of \({\mathbb{C}}^{n+1}\), then any \(C^{\infty}\) CR diffeomorphism of M onto M’ is real-analytic and thus has the holomorphic extension dictated by Hartog’s theorem. Moreover, by using a result of St. Bell and E. Ligocka [Invent. Math. 57, 283-289 (1980; Zbl 0411.32010)] it is shown that if \(\Omega\) and \(\Omega\) ’ are two bounded weakly pseudoconvex domains in \({\mathbb{C}}^{n+1}\), with analytic boundaries, then any biholomorphic diffeomorphism of \(\Omega\) onto \(\Omega\) ’ extends as a biholomorphism between open neighborhoods of their closures.

The present results are closely related to those obtained by K. Diederich and S. M. Webster [Duke Math. J. 47, 835-843 (1980; Zbl 0451.32008)], by C. K. Han [Invent. Math. 73, 51-69 (1983; Zbl 0517.32007)] and by the previously cited Derridy who also considers a class of weakly pseudoconvex domains in \({\mathbb{C}}^ 2\). Moreover, the approach used in the paper allows the author to extend and provide simpler proofs for the previously cited results of Lewy-Pincuk and Webster. The paper also contains results on essential finiteness of real- analytic hypersurfaces in \({\mathbb{C}}^{n+1}\).

The present paper treats the general case in which M and M’ are generic real-analytic submanifolds of \({\mathbb{C}}^{n+d}\) of codimension \(d\geq 1\). The method of proof is based on a version of the edge of the wedge theorem where the standard complex structure is replaced by a new one, which allows the embedding of M in \({\mathbb{C}}^{n+d}\) as \({\mathbb{C}}^ n\times {\mathbb{R}}^ d\), but whose underlying real-analytic structure is the same as the standard one. This procedure resembles one used previously by M. Derridj [Invent. Math. 79, 197-215 (1985; Zbl 0538.32009)] and it enables the authors to use arguments of holomorphic extension from \({\mathbb{R}}^ d\) to \({\mathbb{C}}^ d\). A key in this procedure is an identity that involves the tangential Cauchy-Riemann vector fields and the defining equations of the manifolds. This identity is used to apply the above edge of the wedge theorem and to show that the components of the diffeomorphism under study are analytic. Using these facts the authors are able to show that J is real-analytic in the following cases: when M and M’ are hypersurfaces in \({\mathbb{C}}^{n+1}\) and M’ is essentially finite; when M and M’ are hypersurfaces of finite type in \(C^ 2\); when M and M’ are contained in compact real-analytic subvarieties of \({\mathbb{C}}^{n+1}\). The last assertion is based in part on a result of K. Diederich and J. Fornaess [Ann. Math., II. Ser. 107, 371-384 (1978; Zbl 0378.32014)] and has some consequences about global analyticity. For example, if M and M’ are the real-analytic boundaries of two bounded open subsets \(\Omega\) and \(\Omega\) ’ of \({\mathbb{C}}^{n+1}\), then any \(C^{\infty}\) CR diffeomorphism of M onto M’ is real-analytic and thus has the holomorphic extension dictated by Hartog’s theorem. Moreover, by using a result of St. Bell and E. Ligocka [Invent. Math. 57, 283-289 (1980; Zbl 0411.32010)] it is shown that if \(\Omega\) and \(\Omega\) ’ are two bounded weakly pseudoconvex domains in \({\mathbb{C}}^{n+1}\), with analytic boundaries, then any biholomorphic diffeomorphism of \(\Omega\) onto \(\Omega\) ’ extends as a biholomorphism between open neighborhoods of their closures.

The present results are closely related to those obtained by K. Diederich and S. M. Webster [Duke Math. J. 47, 835-843 (1980; Zbl 0451.32008)], by C. K. Han [Invent. Math. 73, 51-69 (1983; Zbl 0517.32007)] and by the previously cited Derridy who also considers a class of weakly pseudoconvex domains in \({\mathbb{C}}^ 2\). Moreover, the approach used in the paper allows the author to extend and provide simpler proofs for the previously cited results of Lewy-Pincuk and Webster. The paper also contains results on essential finiteness of real- analytic hypersurfaces in \({\mathbb{C}}^{n+1}\).

Reviewer: J.Burbea

##### MSC:

32C05 | Real-analytic manifolds, real-analytic spaces |

32Q99 | Complex manifolds |

32V40 | Real submanifolds in complex manifolds |

32T99 | Pseudoconvex domains |

32D15 | Continuation of analytic objects in several complex variables |