Cauchy-type integrals in several complex variables
We present the theory of Cauchy–Fantappié integral operators, with emphasis on the situation when the domain of integration, D , has minimal boundary regularity. Among these operators we focus on those that are more closely related to the classical Cauchy integral for a planar domain, whose kernel i...
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| Vydané v: | Bulletin of mathematical sciences Ročník 3; číslo 2; s. 241 - 285 |
|---|---|
| Hlavní autori: | , |
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| Jazyk: | English |
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Springer International Publishing
01.08.2013
World Scientific Publishing Co. Pte., Ltd |
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| ISSN: | 1664-3607, 1664-3615 |
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| Abstract | We present the theory of Cauchy–Fantappié integral operators, with emphasis on the situation when the domain of integration,
D
, has minimal boundary regularity. Among these operators we focus on those that are more closely related to the classical Cauchy integral for a planar domain, whose kernel is a holomorphic function of the parameter
z
∈
D
. The goal is to prove
L
p
estimates for these operators and, as a consequence, to obtain
L
p
estimates for the canonical Cauchy–Szegö and Bergman projection operators (which are not of Cauchy–Fantappié type). |
|---|---|
| AbstractList | (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) We present the theory of Cauchy-Fantappié integral operators, with emphasis on the situation when the domain of integration, ..., has minimal boundary regularity. Among these operators we focus on those that are more closely related to the classical Cauchy integral for a planar domain, whose kernel is a holomorphic function of the parameter ... The goal is to prove ... estimates for these operators and, as a consequence, to obtain ... estimates for the canonical Cauchy-Szegö and Bergman projection operators (which are not of Cauchy-Fantappié type). We present the theory of Cauchy–Fantappié integral operators, with emphasis on the situation when the domain of integration, D , has minimal boundary regularity. Among these operators we focus on those that are more closely related to the classical Cauchy integral for a planar domain, whose kernel is a holomorphic function of the parameter z ∈ D . The goal is to prove L p estimates for these operators and, as a consequence, to obtain L p estimates for the canonical Cauchy–Szegö and Bergman projection operators (which are not of Cauchy–Fantappié type). |
| Author | Lanzani, Loredana Stein, Elias M. |
| Author_xml | – sequence: 1 givenname: Loredana surname: Lanzani fullname: Lanzani, Loredana email: loredana.lanzani@gmail.com organization: Department of Mathematics, University of Arkansas – sequence: 2 givenname: Elias M. surname: Stein fullname: Stein, Elias M. organization: Department of Mathematics, Princeton University |
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| References | Lanzani, L., Stein E.M.: The Bergman projection in Lp for domains with minimal smoothness. Illinois J. Math. (to appear) (arXiv:1201.4148) PhongDSteinEMEstimates for the Bergman and Szegö projections on strongly pseudo-convex domainsDuke Math. J.197744369570410.1215/S0012-7094-77-04429-5450623450623, 0392.32014, 10.1215/S0012-7094-77-04429-5 Henkin, G.M.: Integral representations of functions holomorphic in strictly pseudo-convex domains and applications to the ∂¯-problem. Mat. Sb. 82, 300–308 (1970). Engl. Transl.: Math. USSR Sb. 11 (1970) 273–281 KrantzSPelosoMThe Bergman kernel and projection on non-smooth worm domainsHouston J. Math.20083487395024483872448387, 1161.32016 LanzaniLSteinEMCauchy–Szegö and Bergman projections on non-smooth planar domainsJ. Geom. Anal.200414638610.1007/BF0292186620305752030575, 1046.30023, 10.1007/BF02921866 McNealJSteinEMMapping properties of the Bergman projection on convex domains of finite typeDuke Math. J.199473117719910.1215/S0012-7094-94-07307-912572821257282, 0801.32008, 10.1215/S0012-7094-94-07307-9 LigockaEThe Hölder continuity of the Bergman projection and proper holomorphic mappingsStudia Math.19848089107781328781328, 0566.32017 EhsaniDLiebILp-estimates for the Bergman projection on strictly pseudo-convex non-smooth domainsMath. Nachr.200828191692910.1002/mana.20071064924315672431567, 1147.32004, 10.1002/mana.200710649 Folland, G.B., Kohn, J.J.: The Neumann problem for the Cauchy–Riemann complex. In: Ann. Math. Studies, vol. 75. Princeton University Press, Princeton (1972) Henkin, G.: Integral representations of functions holomorphic in strictly pseudo-convex domains and some applications. Mat. Sb. 78, 611–632 (1969). Engl. Transl.: Math. USSR Sb. 7 (1969) 597–616 McNealJEstimates on the Bergman kernel of convex domainsAdv. Math.199410910813910.1006/aima.1994.108213027591302759, 0816.32018, 10.1006/aima.1994.1082 McNealJBoundary behavior of the Bergman kernel function in C2Duke Math. J.198958249951210.1215/S0012-7094-89-05822-510164311016431, 0675.32020, 10.1215/S0012-7094-89-05822-5 HenkinGMLeitererJTheory of Functions on Complex Manifolds1984BaselBirkhäuser0726.32001 BarrettDLanzaniLThe Leray transform on weighted boundary spaces for convex Rembrandt domainsJ. Funct. Anal.20092572780281910.1016/j.jfa.2009.04.01125597172559717, 1181.32002, 10.1016/j.jfa.2009.04.011 CharpentierPDupainYEstimates for the Bergman and Szegő Projections for pseudo-convex domains of finite type with locally diagonalizable Levi formsPubl. Math.20065041344610.5565/PUBLMAT_50206_0822736682273668, 1120.32002 HörmanderLNotions of Convexity1994BaselBirkhäuser0835.32001 Bekollé, D., Bonami, A.: Inegalites a poids pour le noyau de Bergman. C. R. Acad. Sci. Paris Ser. A-B 286(18), A775–A778 (1978) McNealJSteinEMThe Szegö projection on convex domainsMath. Zeit.199722451955310.1007/PL0000459314520481452048, 0948.32004, 10.1007/PL00004593 SteinEMBoundary Behavior of Holomorphic Functions of Several Complex Variables1972PrincetonPrinceton University Press0242.32005 Andersson, M., Passare, M., Sigurdsson, R.: Complex convexity and analytic functionals. Birkhäuser, Basel (2004) HanssonTOn Hardy spaces in complex ellipsoidsAnn. Inst. Fourier (Grenoble)1999491477150110.5802/aif.172717238241723824, 0944.32004, 10.5802/aif.1727 KrantzSFunction theory of several complex variables20012ProvidenceAmerican Mathematical Society1087.32001 HedenmalmHThe dual of a Bergman space on simply connected domainsJ. d’ Analyse20028831133510.1007/BF0278658019797751979775, 1043.46024, 10.1007/BF02786580 ChenS-CShawM-CPartial differential equations in several complex variables2001ProvidenceAmerican Mathematical Society0963.32001 RudinWFunction Theory in the Unit Ball of Cn1980BerlinSpringer10.1007/978-1-4613-8098-60495.32001, 10.1007/978-1-4613-8098-6 BarrettDEBehavior of the Bergman projection on the Diederich-Fornæss wormActa Math.199216811010.1007/BF0239297511498631149863, 0779.32013, 10.1007/BF02392975 DavidGJournéJLSemmesSOprateurs de Caldern-Zygmund, fonctions para-accrtives et interpolationRev. Mat. Iberoamericana19851415610.4171/RMI/17850408850408, 0604.42014, 10.4171/RMI/17 RamirezEEin divisionproblem und randintegraldarstellungen in der komplexen analysisAnn. Math.197018417218710.1007/BF013515610189.09702, 10.1007/BF01351561 BarrettDEIrregularity of the Bergman projection on a smooth bounded domainAnn. Math.198411943143610.2307/20070450566.32016, 10.2307/2007045 KerzmanNSteinEMThe Cauchy–Szegö kernel in terms of the Cauchy–Fantappié kernelsDuke Math. J.19782519722410.1215/S0012-7094-78-04513-1508154508154, 10.1215/S0012-7094-78-04513-1 DavidGOpérateurs intégraux singuliers sur certain courbes du plan complexeAnn. Sci. Éc. Norm. Sup.1984171571890537.42016 NagelARosayJ-PSteinEMWaingerSEstimates for the Bergman and Szegö kernels in C2Ann. Math.1989129211314910.2307/1971487979602979602, 0667.32016, 10.2307/1971487 BellS.LigockaE.A simplification and extension of Fefferman’s theorem on biholomorphic mappingsInvent. Math.1980573283289 BonamiALohouéNProjecteurs de Bergman et Szegő pour une classe de domaines faiblement pseudo-convexes et estimations LpCompositio Math.1982462159226659922659922, 0538.32005 FeffermanCThe Bergman kernel and biholomorphic mappings of pseudo-convex domainsInvent. Math.19742616510.1007/BF01406845350069350069, 0289.32012, 10.1007/BF01406845 Zeytuncu, Y.: Lp-regularity of weighted Bergman projections. Trans. AMS. (2013, to appear) RangeMHolomorphic Functions and Integral Representations in Several Complex Variables1986BerlinSpringer10.1007/978-1-4757-1918-50591.32002, 10.1007/978-1-4757-1918-5 38_CR1 L Lanzani (38_CR24) 2004; 14 38_CR5 38_CR6 L Hörmander (38_CR20) 1994 M Range (38_CR34) 1986 38_CR25 J McNeal (38_CR27) 1989; 58 J McNeal (38_CR29) 1994; 73 P Charpentier (38_CR8) 2006; 50 C Fefferman (38_CR13) 1974; 26 G David (38_CR10) 1984; 17 A Nagel (38_CR31) 1989; 129 DE Barrett (38_CR2) 1984; 119 G David (38_CR11) 1985; 1 D Ehsani (38_CR12) 2008; 281 J McNeal (38_CR30) 1997; 224 38_CR17 S Krantz (38_CR22) 2001 D Phong (38_CR32) 1977; 44 38_CR37 38_CR14 N Kerzman (38_CR21) 1978; 25 S Krantz (38_CR23) 2008; 34 DE Barrett (38_CR3) 1992; 168 E Ligocka (38_CR26) 1984; 80 J McNeal (38_CR28) 1994; 109 A Bonami (38_CR7) 1982; 46 S-C Chen (38_CR9) 2001 38_CR18 GM Henkin (38_CR19) 1984 EM Stein (38_CR36) 1972 W Rudin (38_CR35) 1980 H Hedenmalm (38_CR16) 2002; 88 E Ramirez (38_CR33) 1970; 184 D Barrett (38_CR4) 2009; 257 T Hansson (38_CR15) 1999; 49 |
| References_xml | – reference: EhsaniDLiebILp-estimates for the Bergman projection on strictly pseudo-convex non-smooth domainsMath. Nachr.200828191692910.1002/mana.20071064924315672431567, 1147.32004, 10.1002/mana.200710649 – reference: KrantzSFunction theory of several complex variables20012ProvidenceAmerican Mathematical Society1087.32001 – reference: Folland, G.B., Kohn, J.J.: The Neumann problem for the Cauchy–Riemann complex. In: Ann. Math. Studies, vol. 75. Princeton University Press, Princeton (1972) – reference: Henkin, G.: Integral representations of functions holomorphic in strictly pseudo-convex domains and some applications. Mat. Sb. 78, 611–632 (1969). Engl. Transl.: Math. USSR Sb. 7 (1969) 597–616 – reference: Andersson, M., Passare, M., Sigurdsson, R.: Complex convexity and analytic functionals. Birkhäuser, Basel (2004) – reference: RangeMHolomorphic Functions and Integral Representations in Several Complex Variables1986BerlinSpringer10.1007/978-1-4757-1918-50591.32002, 10.1007/978-1-4757-1918-5 – reference: FeffermanCThe Bergman kernel and biholomorphic mappings of pseudo-convex domainsInvent. Math.19742616510.1007/BF01406845350069350069, 0289.32012, 10.1007/BF01406845 – reference: HenkinGMLeitererJTheory of Functions on Complex Manifolds1984BaselBirkhäuser0726.32001 – reference: ChenS-CShawM-CPartial differential equations in several complex variables2001ProvidenceAmerican Mathematical Society0963.32001 – reference: Bekollé, D., Bonami, A.: Inegalites a poids pour le noyau de Bergman. C. R. Acad. Sci. Paris Ser. A-B 286(18), A775–A778 (1978) – reference: McNealJSteinEMMapping properties of the Bergman projection on convex domains of finite typeDuke Math. J.199473117719910.1215/S0012-7094-94-07307-912572821257282, 0801.32008, 10.1215/S0012-7094-94-07307-9 – reference: RudinWFunction Theory in the Unit Ball of Cn1980BerlinSpringer10.1007/978-1-4613-8098-60495.32001, 10.1007/978-1-4613-8098-6 – reference: BarrettDEIrregularity of the Bergman projection on a smooth bounded domainAnn. Math.198411943143610.2307/20070450566.32016, 10.2307/2007045 – reference: HörmanderLNotions of Convexity1994BaselBirkhäuser0835.32001 – reference: McNealJEstimates on the Bergman kernel of convex domainsAdv. Math.199410910813910.1006/aima.1994.108213027591302759, 0816.32018, 10.1006/aima.1994.1082 – reference: SteinEMBoundary Behavior of Holomorphic Functions of Several Complex Variables1972PrincetonPrinceton University Press0242.32005 – reference: CharpentierPDupainYEstimates for the Bergman and Szegő Projections for pseudo-convex domains of finite type with locally diagonalizable Levi formsPubl. Math.20065041344610.5565/PUBLMAT_50206_0822736682273668, 1120.32002 – reference: LanzaniLSteinEMCauchy–Szegö and Bergman projections on non-smooth planar domainsJ. Geom. Anal.200414638610.1007/BF0292186620305752030575, 1046.30023, 10.1007/BF02921866 – reference: HanssonTOn Hardy spaces in complex ellipsoidsAnn. Inst. Fourier (Grenoble)1999491477150110.5802/aif.172717238241723824, 0944.32004, 10.5802/aif.1727 – reference: McNealJBoundary behavior of the Bergman kernel function in C2Duke Math. J.198958249951210.1215/S0012-7094-89-05822-510164311016431, 0675.32020, 10.1215/S0012-7094-89-05822-5 – reference: HedenmalmHThe dual of a Bergman space on simply connected domainsJ. d’ Analyse20028831133510.1007/BF0278658019797751979775, 1043.46024, 10.1007/BF02786580 – reference: DavidGJournéJLSemmesSOprateurs de Caldern-Zygmund, fonctions para-accrtives et interpolationRev. Mat. Iberoamericana19851415610.4171/RMI/17850408850408, 0604.42014, 10.4171/RMI/17 – reference: Lanzani, L., Stein E.M.: The Bergman projection in Lp for domains with minimal smoothness. Illinois J. Math. (to appear) (arXiv:1201.4148) – reference: NagelARosayJ-PSteinEMWaingerSEstimates for the Bergman and Szegö kernels in C2Ann. Math.1989129211314910.2307/1971487979602979602, 0667.32016, 10.2307/1971487 – reference: PhongDSteinEMEstimates for the Bergman and Szegö projections on strongly pseudo-convex domainsDuke Math. J.197744369570410.1215/S0012-7094-77-04429-5450623450623, 0392.32014, 10.1215/S0012-7094-77-04429-5 – reference: KerzmanNSteinEMThe Cauchy–Szegö kernel in terms of the Cauchy–Fantappié kernelsDuke Math. J.19782519722410.1215/S0012-7094-78-04513-1508154508154, 10.1215/S0012-7094-78-04513-1 – reference: BarrettDEBehavior of the Bergman projection on the Diederich-Fornæss wormActa Math.199216811010.1007/BF0239297511498631149863, 0779.32013, 10.1007/BF02392975 – reference: Zeytuncu, Y.: Lp-regularity of weighted Bergman projections. Trans. AMS. (2013, to appear) – reference: LigockaEThe Hölder continuity of the Bergman projection and proper holomorphic mappingsStudia Math.19848089107781328781328, 0566.32017 – reference: McNealJSteinEMThe Szegö projection on convex domainsMath. Zeit.199722451955310.1007/PL0000459314520481452048, 0948.32004, 10.1007/PL00004593 – reference: RamirezEEin divisionproblem und randintegraldarstellungen in der komplexen analysisAnn. Math.197018417218710.1007/BF013515610189.09702, 10.1007/BF01351561 – reference: BonamiALohouéNProjecteurs de Bergman et Szegő pour une classe de domaines faiblement pseudo-convexes et estimations LpCompositio Math.1982462159226659922659922, 0538.32005 – reference: KrantzSPelosoMThe Bergman kernel and projection on non-smooth worm domainsHouston J. Math.20083487395024483872448387, 1161.32016 – reference: BarrettDLanzaniLThe Leray transform on weighted boundary spaces for convex Rembrandt domainsJ. Funct. Anal.20092572780281910.1016/j.jfa.2009.04.01125597172559717, 1181.32002, 10.1016/j.jfa.2009.04.011 – reference: BellS.LigockaE.A simplification and extension of Fefferman’s theorem on biholomorphic mappingsInvent. Math.1980573283289 – reference: DavidGOpérateurs intégraux singuliers sur certain courbes du plan complexeAnn. Sci. Éc. Norm. Sup.1984171571890537.42016 – reference: Henkin, G.M.: Integral representations of functions holomorphic in strictly pseudo-convex domains and applications to the ∂¯-problem. Mat. Sb. 82, 300–308 (1970). Engl. Transl.: Math. USSR Sb. 11 (1970) 273–281 – volume: 50 start-page: 413 year: 2006 ident: 38_CR8 publication-title: Publ. Math. doi: 10.5565/PUBLMAT_50206_08 – volume: 25 start-page: 197 year: 1978 ident: 38_CR21 publication-title: Duke Math. J. doi: 10.1215/S0012-7094-78-04513-1 – volume: 80 start-page: 89 year: 1984 ident: 38_CR26 publication-title: Studia Math. doi: 10.4064/sm-80-2-89-107 – volume: 224 start-page: 519 year: 1997 ident: 38_CR30 publication-title: Math. Zeit. doi: 10.1007/PL00004593 – volume: 257 start-page: 2780 year: 2009 ident: 38_CR4 publication-title: J. Funct. Anal. doi: 10.1016/j.jfa.2009.04.011 – volume: 17 start-page: 157 year: 1984 ident: 38_CR10 publication-title: Ann. Sci. Éc. Norm. 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