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Condition R and holomorphic mappings of domains with generic corners. (English) Zbl 1307.32015
We say that a domain \(D\subset\mathbb C^n\) satisfies Condition R if the orthogonal Hilbert space projection \(P:L^2(D)\longrightarrow L^2(D)\cap\mathcal O(D)\) is such that \(P(L^2(D)\cap\mathcal C^\infty(\overline D))\subset\mathcal C^\infty(\overline D)\).
We say that a bounded domain \(\varOmega\subset\mathbb C^n\) is a domain with generic corners if \(\varOmega=\bigcup_{j=1}^N\varOmega_j\), where \(\varOmega_1,\dots,\varOmega_N\) are smooth domains, the intersection \(\partial\varOmega_j\) and \(\partial\varOmega_k\) is transversal for all \(j\neq k\), and for each set \(S\subset\{1,\dots,N\}\), if \(B_S:=\bigcap_{j\in S}\partial\varOmega_j\neq\emptyset\), then \(B_S\) is a CR-manifold of dimension \(n-\# S\). Assume that \(D\subset\mathbb C^n\) is a pseudoconvex domain with generic corners which satisfies Condition R. The following two theorems are the main results of the paper.
For any biholomorphic mapping \(f:D\longrightarrow G\) the following conditions are equivalent:
\(f\) extends to a \(\mathcal C^\infty\)-diffeomorphism \(\overline D\longrightarrow\overline G\);
\(G\) is a domain with generic corners and satisfies Condition R.
Let \(G\subset\mathbb C^n\) is a bounded domain of the form \(G=G_1\times\dots\times G_k\), where \(G_1,\dots, G_k\) are smooth. Then every proper holomorphic mapping \(f:D\longrightarrow G\) extends continuously to \(\overline D\) in such a way that the extension is \(\mathcal C^\infty\)-smooth on \(\partial D^{\text{reg}}:=\{z\in\partial D:\) there exists at most one \(j\in\{1,\dots,N\}\) with \(z\in\partial\varOmega_j\}\).

32H40 Boundary regularity of mappings in several complex variables
Full Text: Euclid arXiv
[1] \beginbarticle \bauthor\binitsD. E. \bsnmBarrett, \batitleRegularity of the Bergman projection on domains with transverse symmetries, \bjtitleMath. Ann. \bvolume258 (\byear1981/82), no. \bissue4, page 441-\blpage446. \endbarticle \endbibitem · Zbl 0486.32015
[2] \beginbchapter \bauthor\binitsD. E. \bsnmBarrett, \bctitleDuality between \(A^{\infty}\) and \(A^{-\infty}\) on domains with nondegenerate corners, \bbtitleMultivariable operator theory (Seattle, WA, 1993), \bsertitleContemp. Math., vol. \bseriesno185, \bpublisherAmer. Math. Soc., \blocationProvidence, RI, \byear1995, pp. page 77-\blpage87. \endbchapter \endbibitem
[3] \beginbarticle \bauthor\binitsD. E. \bsnmBarrett and \bauthor\binitsS. \bsnmVassiliadou, \batitleThe Bergman kernel on the intersection of two balls in \(\cx^2\), \bjtitleDuke Math. J. \bvolume120 (\byear2003), no. \bissue2, page 441-\blpage467. \endbarticle \endbibitem · Zbl 1051.32004
[4] \beginbarticle \bauthor\binitsS. R. \bsnmBell, \batitleBiholomorphic mappings and the \(\dbar\)-problem, \bjtitleAnn. of Math. (2) \bvolume114 (\byear1981), no. \bissue1, page 103-\blpage113. \endbarticle \endbibitem · Zbl 0423.32009
[5] \beginbarticle \bauthor\binitsS. R. \bsnmBell, \batitleProper holomorphic mappings and the Bergman projection, \bjtitleDuke Math. J. \bvolume48 (\byear1981), no. \bissue1, page 167-\blpage175. \endbarticle \endbibitem · Zbl 0465.32014
[6] \beginbchapter \bauthor\binitsS. R. \bsnmBell, \bctitleLocal boundary behavior of proper holomorphic mappings, \bbtitleComplex analysis of several variables (Madison, WI, 1982), \bsertitleProc. Sympos. Pure Math., vol. \bseriesno41, \bpublisherAmer. Math. Soc., \blocationProvidence, RI, \byear1984, pp. page 1-\blpage7. \endbchapter \endbibitem
[7] \beginbarticle \bauthor\binitsS. R. \bsnmBell and \bauthor\binitsH. P. \bsnmBoas, \batitleRegularity of the Bergman projection in weakly pseudoconvex domains, \bjtitleMath. Ann. \bvolume257 (\byear1981), page 23-\blpage30. \endbarticle \endbibitem · Zbl 0451.32017
[8] \beginbarticle \bauthor\binitsS. R. \bsnmBell and \bauthor\binitsD. \bsnmCatlin, \batitleBoundary regularity of proper holomorphic mappings, \bjtitleDuke Math. J. \bvolume49 (\byear1982), no. \bissue2, page 385-\blpage396. \endbarticle \endbibitem · Zbl 0475.32011
[9] \beginbarticle \bauthor\binitsS. \bsnmBell and \bauthor\binitsE. \bsnmLigocka, \batitleA simplification and extension of Fefferman’s theorem on biholomorphic mappings, \bjtitleInvent. Math. \bvolume57 (\byear1980), no. \bissue3, page 283-\blpage289. \endbarticle \endbibitem · Zbl 0411.32010
[10] \beginbarticle \bauthor\binitsF. \bsnmBerteloot, continuity of proper holomorphic mappings, \bjtitleStudia Math. \bvolume100 (\byear1991), no. \bissue3, page 229-\blpage235. \endbarticle \endbibitem
[11] \beginbarticle \bauthor\binitsD. \bsnmChakrabarti, \batitleSpectrum of the complex Laplacian on product domains, \bjtitleProc. Amer. Math. Soc. \bvolume138 (\byear2010), no. \bissue9, page 3187-\blpage3202. \endbarticle \endbibitem · Zbl 1225.32039
[12] \beginbarticle \bauthor\binitsD. \bsnmChakrabarti and \bauthor\binitsM.-C. \bsnmShaw, \batitleThe Cauchy-Riemann equations on product domains, \bjtitleMath. Ann. \bvolume349 (\byear2011), no. \bissue4, page 977-\blpage998. \endbarticle \endbibitem · Zbl 1223.32023
[13] \beginbarticle \bauthor\binitsD. \bsnmChakrabarti and \bauthor\binitsK. \bsnmVerma, \batitleCondition R and proper holomorphic maps between equidimensional product domains, \bjtitleAdv. Math. \bvolume248 (\byear2013), page 820-\blpage842. \endbarticle \endbibitem · Zbl 1408.32019
[14] \beginbbook \bauthor\binitsS.-C. \bsnmChen and \bauthor\binitsM.-C. \bsnmShaw, \bbtitlePartial differential equations in several complex variables, \bsertitleAmer. Math. Soc., \blocationProvidence, RI; International Press, Boston, MA, \byear2001. \endbbook \endbibitem
[15] \beginbarticle \bauthor\binitsK. \bsnmDiederich and \bauthor\binitsJ. E. \bsnmFornaess, \batitlePseudoconvex domains: Bounded strictly plurisubharmonic exhaustion functions, \bjtitleInvent. Math. \bvolume39 (\byear1977), no. \bissue2, page 129-\blpage141. \endbarticle \endbibitem · Zbl 0353.32025
[16] \beginbarticle \bauthor\binitsK. \bsnmDiederich and \bauthor\binitsJ. E. \bsnmFornaess, \batitleBoundary regularity of proper holomorphic mappings, \bjtitleInvent. Math. \bvolume67 (\byear1982), page 363-\blpage384. \endbarticle \endbibitem · Zbl 0501.32010
[17] \beginbarticle \bauthor\binitsD. \bsnmEhsani, \batitleSolution of the \(\dbar\)-Neumann problem on a non-smooth domain, \bjtitleIndiana Univ. Math. J. \bvolume52 (\byear2003), no. \bissue3, page 629-\blpage666. \endbarticle \endbibitem · Zbl 1056.32024
[18] \beginbarticle \bauthor\binitsC. \bsnmFefferman, \batitleThe Bergman kernel and biholomorphic mappings of pseudoconvex domains, \bjtitleInvent. Math. \bvolume26 (\byear1974), page 1-\blpage65. \endbarticle \endbibitem · Zbl 0289.32012
[19] \beginbarticle \bauthor\binitsF. , \batitleA reflection principle on strongly pseudoconvex domains with generic corners, \bjtitleMath. Z. \bvolume213 (\byear1993), no. \bissue1, page 49-\blpage64. \endbarticle \endbibitem · Zbl 0790.32024
[20] \beginbarticle \bauthor\binitsP. S. \bsnmHarrington, \batitleThe order of plurisubharmonicity on pseudoconvex domains with Lipschitz boundaries, \bjtitleMath. Res. Lett. \bvolume15 (\byear2008), no. \bissue3, page 485-\blpage490. \endbarticle \endbibitem · Zbl 1146.32016
[21] \beginbbook \bauthor\binitsM. \bsnmJarnicki and \bauthor\binitsP. \bsnmPflug, \bbtitleInvariant distances and metrics in complex analysis, \bpublisherWalter de Gruyter & Co., \blocationBerlin, \byear1993. \endbbook \endbibitem
[22] \beginbbook \bauthor\binitsB. \bsnmMalgrange, \bbtitleLectures on the theory of functions of several complex variables, \bsertitledistributed for the Tata Institute of Fundamental Research, \bpublisherBombay, by Springer-Verlag, \blocationBerlin, \byear1984. \endbbook \endbibitem
[23] \beginbarticle \bauthor\binitsR. M. \bsnmRange, \batitleOn the topological extension to the boundary of biholomorphic maps in \(\mathbb C^n\), \bjtitleTrans. Amer. Math. Soc. \bvolume216 (\byear1976), page 203-\blpage216. \endbarticle \endbibitem
[24] \beginbbook \bauthor\binitsE. J. \bsnmStraube, \bbtitleLectures on the \(\mathcal L^2\)-Sobolev theory of the \(\dbar\)-Neumann problem, \bpublisherEuropean Mathematical Society, , \byear2010. \endbbook \endbibitem · Zbl 1247.32003
[25] \beginbarticle \bauthor\binitsS. \bsnmWebster, \batitleHolomorphic mappings of domains with generic corners, \bjtitleProc. Amer. Math. Soc. \bvolume86 (\byear1982), no. \bissue2, page 236-\blpage240. \endbarticle \endbibitem · Zbl 0505.32014
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