Hardy Spaces and Canonical Kernels on Quadric CR Manifolds

CR functions on an embedded quadric M always extend holomorphically to \(M+i\Gamma _M\) $M+i{\mathrm{\Gamma }}_M$ where \(\Gamma _M\) ${\mathrm{\Gamma }}_M$ is the closure of the convex hull of the image of the Levi form. When \(\Gamma _M\) ${\mathrm{\Gamma }}_M$ is a closed polygonal cone, we show that the Bergman kernel on the interior of \(M+i\Gamma _M\) $M+i{\mathrm{\Gamma }}_M$ is a derivative of the Szegö kernel. Moreover, we develop the \(L^p\) $L^p$ Hardy space theory which turns out to be particularly robust. We provide examples that show that it is unclear how to formulate a corresponding relationship between the Bergman and Szegö kernels on a wider class of quadrics.