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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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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