Cauchy-Leray-Fantappiè integral in linearly convex domains
Let \(\Omega\) be a strictly linearly convex domain with defining function \(\rho\), and let \(dS\) be the measure generated by the form \(\partial\rho\wedge (\overline{\partial}\partial\rho)^{n-1}\). The author proves that the operator \[ K f(z) = \int_{\partial\Omega} \frac{f(\xi)\, dS(\xi)}{\langle\partial\rho(\xi), \xi-z \rangle^n}, \quad z\in\Omega, \] maps \(L^2(\partial\Omega, dS)\) into the Hardy space \(H^2(\Omega)\) defined with respect to \(dS\). Also, he shows that \(K\) and \(K^*\) are bounded on \(\mathrm{BMO}(\partial\Omega)\). The proofs are based on ideas of \textit{T. Hansson} [Ann. Inst. Fourier 49, No. 5, 1477--1501 (1999; Zbl 0944.32004)].
scientific article; zbMATH DE number 6347715