Duke Mathematical Journal

The Cauchy–Szegő projection for domains in Cn with minimal smoothness

Loredana Lanzani and Elias M. Stein

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Abstract

We prove the Lp(bD)-regularity of the Cauchy–Szegő projection (also known as the Szegő projection) for bounded domains DCn which are strongly pseudoconvex and whose boundary satisfies the minimal regularity condition of class C2.

Article information

Source
Duke Math. J., Volume 166, Number 1 (2017), 125-176.

Dates
Received: 12 June 2015
Revised: 18 February 2016
First available in Project Euclid: 12 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1478919691

Digital Object Identifier
doi:10.1215/00127094-3714757

Mathematical Reviews number (MathSciNet)
MR3592690

Zentralblatt MATH identifier
1367.32005

Subjects
Primary: 30E20: Integration, integrals of Cauchy type, integral representations of analytic functions [See also 45Exx] 32A50: Harmonic analysis of several complex variables [See mainly 43-XX] 32A55: Singular integrals 32A25: Integral representations; canonical kernels (Szego, Bergman, etc.)
Secondary: 31A10: Integral representations, integral operators, integral equations methods 32A26: Integral representations, constructed kernels (e.g. Cauchy, Fantappiè- type kernels) 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 46E22: Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32] 47B34: Kernel operators 31B10: Integral representations, integral operators, integral equations methods

Keywords
Cauchy integral $T(1)$ theorem space of homogeneous type Leray–Levi measure Cauchy–Szegő projection Hardy space Lebesgue space pseudoconvex domain minimal smoothness

Citation

Lanzani, Loredana; Stein, Elias M. The Cauchy–Szegő projection for domains in $\mathbb{C}^{n}$ with minimal smoothness. Duke Math. J. 166 (2017), no. 1, 125--176. doi:10.1215/00127094-3714757. https://projecteuclid.org/euclid.dmj/1478919691


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References

  • [1] P. Ahern and R. Schneider, A smoothing property of the Henkin and Szegő projections, Duke Math. J. 47 (1980), 135–143.
  • [2] D. E. Barrett, Irregularity of the Bergman projection on a smooth bounded domain in $\mathbf{C}^{2}$, Ann. of Math. (2) 119 (1984), 431–436.
  • [3] A. Bonami and N. Lohoué, Projecteurs de Bergman et Szegő pour une classe de domaines faiblement pseudo-convexes et estimations $L^{p}$, Compos. Math. 46, (1982), 159–226.
  • [4] A.-P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Natl. Acad. Sci. USA 74 (1977), 1324–1327.
  • [5] P. Charpentier and Y. Dupain, Estimates for the Bergman and Szegő projections for pseudoconvex domains of finite type with locally diagonalizable Levi form, Publ. Mat. 50 (2006), 413–446.
  • [6] S.-C. Chen and M.-C. Shaw, Partial Differential Equations in Several Complex Variables, Amer. Math. Soc., Providence, 2001.
  • [7] M. Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), 601–628.
  • [8] M. Christ, Lectures on Singular Integral Operators, CBMS Reg. Conf. Ser. Math. 77, Amer. Math. Soc., Providence, 1990.
  • [9] R. R. Coifman, A. McIntosh, and Y. Meyer, L’intègrale de Cauchy dèfinit un opèrateur bornè sur $L^{2}$ pour les courbes lipschitziennes, Ann. of Math. (2) 116 (1982), 361–387.
  • [10] A. Cumenge, Comparaison des projecteurs de Bergman et Szegő et applications, Ark. Mat. 28 (1990), 23–47.
  • [11] G. David, J. L. Journé, and S. Semmes, Opérateurs de Calderòn-Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoam. 1 (1985), 1–56.
  • [12] C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1–65.
  • [13] G. B. Folland and J. J. Kohn, The Neumann Problem for the Cauchy-Riemann Complex, Ann. of Math. Stud. 75, Princeton Univ. Press, Princeton, 1972.
  • [14] G. Francsics and N. Hanges, Explicit formulas for the Szegő kernel on certain weakly pseudoconvex domains, Proc. Amer. Math. Soc. 123 (1995), 3161–3168.
  • [15] G. Francsics and N. Hanges, Trèves curves and the Szegő kernel, Indiana Univ. Math. J. 47 (1998), 995–1009.
  • [16] J. Halfpap, A. Nagel, and S. Wainger, The Bergman and Szegő kernels near points of infinite type, Pacific J. Math. 246 (2010), 75–128.
  • [17] N. Hanges, Explicit formulas for the Szegő kernel for some domains in $\mathbb{C}^{2}$, J. Funct. Anal. 88 (1990), 153–165.
  • [18] N. Kerzman and E. M. Stein, The Cauchy kernel, the Szegő kernel and the Riemann mapping function, Math. Ann. 236 (1978), 85–93.
  • [19] N. Kerzman and E. M. Stein, The Szegő kernel in terms of the Cauchy-Fantappiè kernels, Duke Math. J. 45 (1978), 197–224.
  • [20] K. D. Koenig, Comparing the Bergman and Szegő projections on domains with subelliptic boundary Laplacian, Math. Ann. 339 (2007), 667–693.
  • [21] K. D. Koenig, An analogue of the Kerzman-Stein formula for the Bergman and Szegő projections, J. Geom. Anal. 19 (2009), 81–86.
  • [22] K. D. Koenig and L. Lanzani, Bergman versus Szegő via conformal mapping, Indiana Univ. Math. J. 58 (2009), 969–997.
  • [23] L. Lanzani and E. M. Stein, Szegő and Bergman projections on non-smooth planar domains, J. Geom. Anal. 14 (2004), 63–86.
  • [24] L. Lanzani and E. M. Stein, The Bergman projection in $L^{p}$ for domains with minimal smoothness, Illinois J. Math. 56 (2012), 127–154.
  • [25] L. Lanzani and E. M. Stein, Cauchy-type integrals in several complex variables, Bull. Math. Sci. 3 (2013), 241–285.
  • [26] L. Lanzani and E. M. Stein, The Cauchy integral in $\mathbb{C}^{n}$ for domains with minimal smoothness, Adv. Math. 264 (2014), 776–830.
  • [27] L. Lanzani and E. M. Stein, “Hardy spaces of holomorphic functions for domains in $\mathbb{C}^{n}$ with minimal smoothness” in Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory (Vol. 1): Celebrating Cora Sadosky’s Life, Assoc. for Women in Math. Ser. 4, Springer, New York, 2016, 179–199.
  • [28] J. D. McNeal and E. M. Stein, The Szegő projection on convex domains, Math. Z. 224 (1997), 519–553.
  • [29] A. Nagel, J.-P. Rosay, E. M. Stein, and S. Wainger, Estimates for the Bergman and Szegő kernels in $\mathbb{C}^{2}$, Ann. of Math. (2) 129 (1989), 113–149.
  • [30] D. H. Phong and E. M. Stein, Estimates for the Bergman and Szegő projections on strongly pseudo-convex domains, Duke Math. J. 44 (1977), 695–704.
  • [31] E. Ramirez de Arellano, Ein Divisionproblem und Randintegraldarstellungen in der komplexen Analysis, Math. Ann. 184 (1969/1970), 172–187.
  • [32] R. M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Grad. Texts in Math. 108, Springer, Berlin, 1986.
  • [33] A. A. Solovév, Estimates in $L^{p}$ of the integral operators that are connected with spaces of analytic and harmonic functions, Dokl. Akad. Nauk SSSR 240 (1978), 1301–1304.