The Szegő kernel as a singular integral kernel on a family of weakly pseudoconvex domains
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- by Katharine Perkins Diaz
- Trans. Amer. Math. Soc. 304 (1987), 141-170
- DOI: https://doi.org/10.1090/S0002-9947-1987-0906810-4
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Abstract:
The Szegö kernels on the weakly pseudoconvex domains $\{ \operatorname {Im} {z_2} > |{z_1}{|^{2k}}\}$, $k \in {Z^ + }$, 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.References
- Patrick Ahern and Robert Schneider, Holomorphic Lipschitz functions in pseudoconvex domains, Amer. J. Math. 101 (1979), no. 3, 543–565. MR 533190, DOI 10.2307/2373797
- Wolfgang Alt, Singuläre Integrale mit gemischten Homogenitäten auf Mannigfaltigkeiten und Anwendungen in der Funktionentheorie, Math. Z. 137 (1974), 227–256. MR 404691, DOI 10.1007/BF01237392
- Aline Bonami and Noël Lohoué, Projecteurs de Bergman et Szegő pour une classe de domaines faiblement pseudo-convexes et estimations $L^{p}$, Compositio Math. 46 (1982), no. 2, 159–226 (French). MR 659922
- Ronald R. Coifman and Guido Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, Berlin-New York, 1971 (French). Étude de certaines intégrales singulières. MR 0499948, DOI 10.1007/BFb0058946
- Katharine Perkins Diaz, The Szegő kernel as a singular integral kernel on a family of weakly pseudoconvex domains, Trans. Amer. Math. Soc. 304 (1987), no. 1, 141–170. MR 906810, DOI 10.1090/S0002-9947-1987-0906810-4
- G. B. Folland and E. M. Stein, Estimates for the $\bar \partial _{b}$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429–522. MR 367477, DOI 10.1002/cpa.3160270403
- Roe W. Goodman, Nilpotent Lie groups: structure and applications to analysis, Lecture Notes in Mathematics, Vol. 562, Springer-Verlag, Berlin-New York, 1976. MR 0442149, DOI 10.1007/BFb0087594
- P. C. Greiner and E. M. Stein, On the solvability of some differential operators of type $cm_{b}$, Several complex variables (Cortona, 1976/1977) Scuola Norm. Sup. Pisa, Pisa, 1978, pp. 106–165. MR 681306
- J. J. Kohn, Boundary behavior of $\delta$ on weakly pseudo-convex manifolds of dimension two, J. Differential Geometry 6 (1972), 523–542. MR 322365
- A. Korányi and S. Vági, Singular integrals on homogeneous spaces and some problems of classical analysis, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 25 (1971), 575–648 (1972). MR 463513
- Steven G. Krantz, Function theory of several complex variables, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR 635928
- Alexander Nagel and E. M. Stein, Lectures on pseudodifferential operators: regularity theorems and applications to nonelliptic problems, Mathematical Notes, vol. 24, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1979. MR 549321
- Alexander Nagel, Elias M. Stein, and Stephen Wainger, Boundary behavior of functions holomorphic in domains of finite type, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), no. 11, 6596–6599. MR 634936, DOI 10.1073/pnas.78.11.6596
- Alexander Nagel, Elias M. Stein, and Stephen Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103–147. MR 793239, DOI 10.1007/BF02392539
- D. H. Phong and E. M. Stein, Estimates for the Bergman and Szegö projections on strongly pseudo-convex domains, Duke Math. J. 44 (1977), no. 3, 695–704. MR 450623, DOI 10.1215/S0012-7094-77-04429-5
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- E. M. Stein, Boundary behavior of holomorphic functions of several complex variables, Mathematical Notes, No. 11, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. MR 0473215
Bibliographic Information
- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 304 (1987), 141-170
- MSC: Primary 32A35; Secondary 32F15, 32H10
- DOI: https://doi.org/10.1090/S0002-9947-1987-0906810-4
- MathSciNet review: 906810