## On the Singularities of Non-Analytic SzegÃ¶ Kernels

J. Math. Sci. Univ. Tokyo
Vol. 6 (1999), No. 1, Page 13--39.

Kamimoto, Joe
On the Singularities of Non-Analytic SzegÃ¶ Kernels
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Abstract:
The \,\, {\it CR} \,\, manifold \,\, $M_m=\,\, \{(z_1,z_2) \, \in \, {\mathbb C}^2 ; \,\, \Im z_2 = [\Re z_1]^{2m} \} (m=2,3,\ldots)$ is a counterexample, which was given by Christ and Geller, to analytic hypoellipticity of $\bar{\partial}_{b}$ and real analyticity of the SzegÃ¶ kernel. In order to give a direct interpretation for the breakdown of real analyticity of the SzegÃ¶ kernel, we give a Borel summation type representation of the SzegÃ¶ kernel in terms of simple singular solutions of the equation $\bar{\partial}_{b}u = 0$.

Keywords: SzegÃ¶ kernel, Bergman kernel, weakly pseudoconvex, of finite type, analytically hypoelliptic, CR manifold, Borel sum

Mathematics Subject Classification (1991): 32C16, 32F20, 32H10
Mathematical Reviews Number: MR1683305

Received: 1998-07-17