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

Diaz, Katharine Perkins

doi:10.1090/S0002-9947-1987-0906810-4

The Szegő kernel as a singular integral kernel on a family of weakly pseudoconvex domains
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Trans. Amer. Math. Soc. 304 (1987), 141-170 Request permission

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.

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