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On Regularity and Irregularity of Certain Holomorphic Singular Integral Operators

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Geometric Aspects of Harmonic Analysis

Part of the book series: Springer INdAM Series ((SINDAMS,volume 45))

Abstract

We survey recent work and announce new results concerning two singular integral operators whose kernels are holomorphic functions of the output variable, specifically the Cauchy–Leray integral and the Cauchy–Szegő projection associated to various classes of bounded domains in \(\mathbb C^n\) with n ≥ 2.

Dedicated to Fulvio Ricci, on the occasion of his 70th birthday

The author Loredana Lanzanis was supported in part by the National Science Foundation, award DMS-1503612.

The author Elias M. Stein was supported in part by the National Science Foundation, award DMS-1700180.

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Notes

  1. 1.

    “Canonical” in the sense that it is the restriction to bD of a universal kernel defined in \(\mathbb C^n\times \mathbb C^n\setminus \{z=w\}\).

  2. 2.

    It is failure of this property that renders the Bochner–Martinelli integral unsuitable for the analysis of \(\mathcal S\).

  3. 3.

    The domain is pseudoconvex but cannot be “exhausted” by smooth pseudoconvex “super-domains”.

  4. 4.

    That is, the orthogonal projection of L 2(D, dV ) onto the Bergman space A 2(D) := 𝜗(D) ∩ L 2(D, dV ).

References

  1. Ahern, P., Schneider, R.: A smoothing property of the Henkin and Cauchy–Szegő projections. Duke Math. J. 47, 135–143 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ahern, P., Schneider, R.: The boundary behavior of Henkin’s kernel. Pac. J. Math. 66, 9–14 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  3. Andersson, M., Passare, M., Sigurdsson, R.: Complex Convexity and Analytic Functionals Birkhäuser, Boston (2004)

    Google Scholar 

  4. Barrett, D.: Behavior of the Bergman projection on the Diederich-Fornæss worm. Acta Math. 168, 1–10 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barrett, D.E., Ehsani, D., Peloso, M.: Regularity of projection operators attached to worm domains. Doc. Math. 20, 1207–1225 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Barrett, D., Lanzani, L.: The spectrum of the Leray transform for convex Reinhardt domains in \(\mathbb C^2\). J. Funct. Anal. 257, 2780–2819 (2009)

    Google Scholar 

  7. Barrett, D.E., Vassiliadou, S.: The Bergman kernel on the intersection of two balls in \(\mathbb C^2\). Duke Math. J. 120, 441–467 (2003)

    Google Scholar 

  8. Bonami, A., Charpentier, P.: Comparing the Bergman and Cauchy–Szegő projections. Math. Z. 204, 225–233 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bonami, A., Lohoué, N.: Projecteurs de Bergman et Cauchy–Szegő pour une classe de domaines faiblement pseudo-convexes et estimations L p. Compos. Math. 46(2), 159–226 (1982)

    MATH  Google Scholar 

  10. Boutet de Monvel, L., Sjöstrand J.: Sur la singularité des noyaux de Bergman et the Szegő, Journées équations aux dérivées partialles de Rennes (1975), pp. 123–164. Astérisque, no. 34–35. Soc. Math. France, Paris (1976)

    Google Scholar 

  11. Calderón, A.: Cauchy integrals on Lipschitz curves and related operators. Proc. Nat. Acad. Sci. U.S.A. 74, 1324–1327 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chakrabarti, D., Zeytuncu, Y.: L p mapping properties of the Bergman projection on the Hartogs triangle. Proc. AMS 144(4), 1643–1653 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Charpentier, P., Dupain, Y.: Estimates for the Bergman and Cauchy–Szegő projections for pseudoconvex domains of finite type with locally diagonalizable Levi forms. Publ. Mat. 50, 413–446 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, L.W., Zeytuncu, Y.: Weighted Bergman projections on the Hartogs triangle: exponential decay. New York J. Math. 22(16), 1271–1282 (2016)

    MathSciNet  MATH  Google Scholar 

  15. Christ, M.: A T(b) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 60/61(2), 601–628 (1990)

    Google Scholar 

  16. Christ, M.: Lectures on Singular Integral Operators, vol. 77. AMS-CBMS (1990)

    Google Scholar 

  17. Coifman, R.R., McIntosh, A., Meyer, Y.: L’intégrale de Cauchy définit un opérateur borné sur L 2 pour les courbes Lipschitziennes. Ann. Math. 116, 361–387 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cumenge, A.: Comparaison des projecteurs de Bergman et Cauchy–Szegő et applications. Ark. Mat. 28, 23–47 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  19. David, G.: Opérateurs intégraux singuliers sur certain courbes du plan complexe. Ann. Sci. École Norm. Sup. 17, 157–189 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  20. Duong, X.-T., Lacey, M., Li, J., Wick, B., Wu, Q.: Commutators of Cauchy-type integrals for domains in \(\mathbb C^n\) with minimal smoothness. Preprint (2018) (ArXiv: 1809.08335)

    Google Scholar 

  21. Duren, P.L.: Theory of H p Spaces. Dover, New York (2000)

    Google Scholar 

  22. Fefferman, C.: The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math. 26, 1–65 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fefferman, C.: Parabolic invariant theory in complex analysis. Adv. Math. 31, 131–162 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gupta, P.: Lower-dimensional Fefferman measures via the Bergman kernel. Contemp. Math. 681, 137–151 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hansson, T.: On Hardy spaces in complex ellipsoids. Ann. Inst. Fourier (Grenoble) 49, 1477–1501 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Henkin, G.M.: Integral representation of functions which are holomorphic in strictly pseudoconvex regions, and some applications (Russian). Mat. Sb. (N.S.) 78, 611–632 (1969)

    Google Scholar 

  27. Hörmander, L.: Notions of Convexity. Progress in Mathematics, vol. 127. Birkhäuser, Boston (1994)

    Google Scholar 

  28. Kenig, C.: Weighted H p spaces on Lipschitz domains. Am. J. Math. 102, 129–163 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kerzman, N., Stein, E.M.: The Cauchy–Szegő kernel in terms of Cauchy–Fantappié kernels. Duke Math. J. 45, 197–224 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kerzman, N., Stein, E.M.: The Cauchy kernel, the Cauchy–Szegő kernel and the Riemann mapping function. Math. Ann. 236, 85–93 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kiselman, C.: A study of the Bergman projection in certain Hartogs domains, in Several Complex Variables and Complex Geometry, Part 3, Proc. Sympos. Pure Math, vol. 52, Part 3, Amer. Math. Soc., Providence (1991)

    Google Scholar 

  32. Kœnig, K.D.: Comparing the Bergman and Cauchy–Szegő projections on domains with subelliptic boundary Laplacian. Math. Ann. 339, 667–693 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kœnig, K.D.: An analogue of the Kerzman–Stein formula for the Bergman and Cauchy–Szegő projections. J. Geom. Anal. 14, 63–86 (2004)

    MathSciNet  Google Scholar 

  34. Kœnig, K.D., Lanzani, L.: Bergman vs. Szegö via conformal mapping. Indiana U. Math. J. 58, 969–997 (2009)

    MATH  Google Scholar 

  35. Krantz, S.G.: Integral formulas in complex analysis, in Bejing Lectures in Harmonic Analysis, Ann. of Math. Stud., vol. 112, pp. 185–240 (1987)

    Google Scholar 

  36. Krantz, S.G.: Function Theory of Several Complex Variables. Wiley, London (1982)

    MATH  Google Scholar 

  37. Krantz, S., Peloso, M.: The Bergman kernel and projection on non-smooth worm domains. Houst. J. Math. 34, 873–950 (2008)

    MathSciNet  MATH  Google Scholar 

  38. Kytmanov, A.M.: The Bochner–Martinelli Integral and Its Applications. Birkhäuser, Basel (1992)

    Google Scholar 

  39. Lanzani, L.: Cauchy–Szegő projection versus potential theory for non-smooth planar domains. Indiana Univ. Math. J. 48, 537–556 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  40. Lanzani, L.: Cauchy transform and Hardy spaces for rough planar domains. Contemp. Math. 251, 409–428 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  41. Lanzani, L.: Harmonic analysis techniques in several complex variables. Bruno Pini Math. Anal. Seminar Ser. 1, 83–110 (2014). ISSN: 2240-2829

    MathSciNet  MATH  Google Scholar 

  42. Lanzani, L., Stein, E.M.: Cauchy–Szegő and Bergman projections on non-smooth planar domains. J. Geom. Anal. 14, 63–86 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  43. Lanzani, L., Stein, E.M.: The Bergman projection in L p for domains with minimal smoothness. Illinois J. Math. 56(1), 127–154 (2013)

    MathSciNet  MATH  Google Scholar 

  44. Lanzani, L., Stein, E.M.: Cauchy-type integrals in several complex variables. Bull. Math. Sci. 3(2), 241–285 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  45. Lanzani, L., Stein, E.M.: The Cauchy integral in \(\mathbb {C}^n\) for domains with minimal smoothness. Adv. Math. 264, 776–830 (2014)

    Google Scholar 

  46. Lanzani, L., Stein, E.M.: The Cauchy–Leray integral: counter-examples to the L p-theory. Indiana U. Math. J. 68(5), 1609–1621 (2019)

    Article  MATH  Google Scholar 

  47. Lanzani, L., Stein, E.M.: The role of an integration identity in the analysis of the Cauchy–Leray transform. Sci. China Math. 60, 1923–1936 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  48. Lanzani, L., Stein, E.M.: The Cauchy–Szegő projection for domains with minimal smoothness. Duke Math. J. 166(1), 125–176 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  49. Lanzani, L., Stein, E.M.: Hardy Spaces of Holomorphic functions for domains in \(\mathbb {C}^n\) with minimal smoothness. In: Harmonic Analysis, Partial Differential Equations, Complex Analysis, and Operator Theory: Celebrating Cora Sadosky’s Life, AWM-Springer vol. 1, pp. 179–200 (2016). ISBN-10:3319309595.

    Google Scholar 

  50. Lanzani, L., Stein, E.M.: On irregularity of the Cauchy–Szegő projection for the Diederich-Fornæss worm domain. Manuscript in preparation

    Google Scholar 

  51. McNeal, J.: Boundary behavior of the Bergman kernel function in \(\mathbb C^2\). Duke Math. J. 58(2), 499–512 (1989)

    Google Scholar 

  52. McNeal, J.: Estimates on the Bergman kernel of convex domains. Adv. Math. 109, 108–139 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  53. McNeal, J., Stein, E.M.: Mapping properties of the Bergman projection on convex domains of finite type. Duke Math. J. 73(1), 177–199 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  54. Meyer, Y., Coifman, R.: Ondelettes et Opérateurs III, Opérateurs multilinéaires, Actualités Mathématiques, Hermann (Paris) (1991), pp. i–xii and 383–538. ISBN: 2-7056-6127-1

    Google Scholar 

  55. Monguzzi, A.: Hardy spaces and the Cauchy–Szegő projection of the non-smooth worm domain \(D^{\prime }_\beta \). J. Math. Anal. Appl. 436, 439–466 (2016)

    Google Scholar 

  56. Monguzzi, A.: On Hardy spaces on worm domains. Concr. Oper. 3, 29–42 (2016)

    MathSciNet  MATH  Google Scholar 

  57. Monguzzi, A., Peloso, M.: Sharp estimates for the Cauchy–Szegő projection on the distinguished boundary of model worm domains. Integr. Equ. Oper. Theory 89, 315–344 (2017)

    Article  MATH  Google Scholar 

  58. Monguzzi, A., Peloso, M.: Regularity of the Cauchy–Szegő projection on model worm domains. Complex Var. Elliptic Equ. 62(9), 1287–1313 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  59. Munasinghe, S., Zeytuncu, Y.: Irregularity of the Szegő projection on bounded pseudoconvex domains in \(\mathbb C^2\). Integr. Equ. Oper. Theory 82, 417–422 (2015)

    MATH  Google Scholar 

  60. Munasinghe, S., Zeytuncu, Y.: L p-regularity of weighted Cauchy–Szegő projections on the unit disc. Pac. J. Math. 276, 449–458 (2015)

    Article  MATH  Google Scholar 

  61. Nagel, A., Pramanik, M.: Diagonal estimates for the Bergman kernel on certain domains in \(\mathbb {C}^n\) (2011). Preprint

    Google Scholar 

  62. Nagel, A., Rosay, J.-P., Stein, E.M., Wainger, S.: Estimates for the Bergman and Cauchy–Szegő kernels in \(\mathbb C^2\). Ann. Math. 129(2), 113–149 (1989)

    Google Scholar 

  63. Nazarov, F., Treil, S., Volberg, A.: Cauchy integral and Calderón–Zygmund operators on nonhomogeneous spaces. Int. Math. Res. Not. 15, 703–726 (1997)

    Article  MATH  Google Scholar 

  64. Phong, D., Stein, E.M.: Estimates for the Bergman and Cauchy–Szegő projections on strongly pseudoconvex domains. Duke Math. J. 44(3), 695–704 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  65. Poletsky, E., Stessin, M.: Hardy and Bergman spaces on hyperconvex domains and their composition operators. Indiana Univ. Math. J. 57, 2153–2201 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  66. Range, R.M.: Holomorphic Functions and Integral Representations in Several Complex Variables. Graduate Texts in Mathematics, vol. 108. Springer, Berlin (1986)

    Google Scholar 

  67. Ramírez de Arellano, E.: Ein Divisionsproblem und Randintegraldarstellungen in der komplexen analysis. Math. Ann. 184, 172–187 (1969/1970)

    Google Scholar 

  68. Rotkevich, A.S.: Cauchy–Leray-Fantappiè integral in linearly convex domains. J. of Math. Sci. 194, 693–702 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  69. Rotkevich, A.S.: The Aizenberg formula in non convex domains and some of its applications. Zap. Nauchn. Semin. POMI 389, 206–231 (2011)

    Google Scholar 

  70. Semmes, S.: The Cauchy integral and related operators on smooth curves. Thesis, Washington University (1983)

    Google Scholar 

  71. Stein, E.M.: Boundary Behavior of Holomorphic Functions of Several Complex Variables. Princeton University Press, Princeton (1972)

    MATH  Google Scholar 

  72. Stein, E.M.: Harmonic Analysis Princeton University Press, Princeton (1993)

    Google Scholar 

  73. Stout, E.L.: H p functions on strictly pseudoconvex domains. Am. J. Math. 98, 821–852 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  74. Tolsa, X.: Analytic capacity, rectifiability, and the Cauchy integral, International Congress of Mathematicians, vol. II, 1505–1527. Eur. Math. Soc., Zürich (2006)

    Google Scholar 

  75. Tolsa, X.: The semiadditivity of continuous analytic capacity and the inner boundary conjecture. Am. J. Math. 126, 523–567 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  76. Verdera, J.: L 2 boundedness of the Cauchy integral and Menger curvature. In: Harmonic Analysis and Boundary Value Problems. Contemp. Math., vol. 277, pp. 139–158. Amer. Math. Soc., Providence (2001)

    Google Scholar 

  77. Zeytuncu, Y.: L p-regularity of weighted Bergman projections. Trans. Am. Math. Soc. 365, 2959–2976 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Syracuse University, Syracuse, NY, USA

    Loredana Lanzani

  2. Department of Mathematics, Princeton University, Princeton, NJ, USA

    Elias M. Stein

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  1. Loredana Lanzani
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Correspondence to Loredana Lanzani .

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  1. Dipartimento di Ingegneria Civile, Edile e Ambientale - ICEA, University of Padova, Padova, Italy

    Paolo Ciatti

  2. School of Mathematics, University of Birmingham, Birmingham, UK

    Alessio Martini

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Lanzani, L., Stein, E.M. (2021). On Regularity and Irregularity of Certain Holomorphic Singular Integral Operators. In: Ciatti, P., Martini, A. (eds) Geometric Aspects of Harmonic Analysis. Springer INdAM Series, vol 45. Springer, Cham. https://doi.org/10.1007/978-3-030-72058-2_14

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