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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The Szegő kernel as a singular integral kernel on a family of weakly pseudoconvex domains


Author: Katharine Perkins Diaz
Journal: Trans. Amer. Math. Soc. 304 (1987), 141-170
MSC: Primary 32A35; Secondary 32F15, 32H10
MathSciNet review: 906810
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Abstract: The Szegö kernels on the weakly pseudoconvex domains $ \{ \operatorname{Im} {z_2} > \vert{z_1}{\vert^{2k}}\} $, $ k \in {Z^ + }$, have been computed by Greiner and Stein. After constructing a global, nonisotropic pseudometric suitable for Calderón-Zygmund singular integral theory on the boundaries of the domains, we study principal value operators associated to these Szegö kernels. We prove that the principal value operators are bounded on $ {L^p}$ for $ 1 < p < \infty $, and that they preserve certain nonisotropic Lipschitz classes. We then derive a Plemelj formula that relates the principal value operators to the Szegö projections. From this formula we deduce that the Szegö projections are also bounded on $ {L^p}$, for $ 1 < p < \infty $, and that they preserve the same nonisotropic Lipschitz classes.


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  • [AhS] P. Ahern and R. Schneider, Holomorphic Lipschitz functions in pseudoconvex domains, Amer. J. Math. 101 (1979), 543-565. MR 533190 (81f:32022)
  • [A1] W. Alt, Singuläre Integrale mit gemischten Homogenitäten auf Mannigfaltigkeiten und Andwendungen in der Funktionentheorie, Math. Z. 137 (1974), 227-256. MR 0404691 (53:8491)
  • [BL] A. Bonami and N. Lohoué, Projecteurs de Bergman et Szegö pour une classe de domaines faiblement pseudo-convexes et estimations $ {L^p}$, Compositio Math., 46 (1982), 159-226. MR 659922 (84b:32008)
  • [CW] R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogenes, Lecture Notes in Math., vol. 242, Springer-Verlag, 1971. MR 0499948 (58:17690)
  • [Di] K. P. Diaz, The Szegö kernel as a singular integral kernel on a family of weakly pseudoconvex domains, Ph.D. dissertation, Princeton Univ., Princeton, N. J., 1986. MR 906810 (89d:32009)
  • [FS] G. B. Folland and E. M. Stein, Estimates for the $ {\overline \partial _b}$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429-522. MR 0367477 (51:3719)
  • [Go] R. W. Goodman, Nilpotent Lie groups: structure and applications to analysis, Lecture Notes in Math., vol. 562 Springer-Verlag, 1976. MR 0442149 (56:537)
  • [GS] P. C. Greiner and E. M. Stein, On the solvability of some differential operators of type $ {\square _b}$, Proc. Internat. Conf., (Cortona, Italy, 1976-1977), Scuola Norm. Sup. Pisa, Pisa, 1978, pp. 106-165. MR 681306 (84d:35111)
  • [Ko] J. J. Kohn, Boundary behavior of $ \overline \partial $ on weakly pseudo-convex manifolds of dimension two, J. Differential Geometry 6 (1972), 523-542. MR 0322365 (48:727)
  • [KoV] A. Korányi and S. Vági, Singular integrals on homogeneous spaces and some problems of classical analysis, Ann. Scuola Norm. Sup. Pisa 25 (1971), 575-648. MR 0463513 (57:3462)
  • [Kr] S. G. Krantz, Function theory of several complex variables, Wiley, 1982. MR 635928 (84c:32001)
  • [NS] A. Nagel and E. M. Stein, Lectures on pseudo-differential operators: regularity theorems and applications to non-elliptic problems, Princeton Univ. Press, Princeton, N. J., 1979. MR 549321 (82f:47059)
  • [NSW1] A. Nagel, E. M. Stein and S. Wainger, Boundary behavior of functions holomorphic in domains of finite type, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), 6596-6599. MR 634936 (82k:32027)
  • [NSW2] -, Balls and metrics defined by vector fields I: Basic properties, Acta Math. 155 (1985), 102-147. MR 793239 (86k:46049)
  • [PS] D. H. Phong and E. M. Stein, Estimates for the Bergman and Szegö projections on strongly pseudo-convex domains, Duke Math. J. 44 (1977), 695-704. MR 0450623 (56:8916)
  • [SI] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N. J., 1979. MR 0290095 (44:7280)
  • [S2] -, Boundary behavior of holomorphic functions of several complex variables, Princeton Univ. Press, Princeton, N. J., 1972. MR 0473215 (57:12890)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0906810-4
PII: S 0002-9947(1987)0906810-4
Article copyright: © Copyright 1987 American Mathematical Society