On certain integrals associated to CR-functions

Abstract:

We construct explicit $\bar \partial$-primitives of Cauchy-Fantappiè type kernels on analytic subvarieties of domains in ${{\mathbf {C}}^n}$, outside the zero set of a holomorphic map defined on the variety. Then we use these primitives to derive, by means of a process of passing to a residue, integral formulas for ${\text {CR}}$-functions defined on subsets of the boundary of a variety. Certain geometric restrictions on the varieties are imposed. These primitives apply in the particular case of the Bochner-Martinelli kernel in domains in ${{\mathbf {C}}^n}$ and we use these primitives, in this case, to prove a criterion for holomorphic extendability of ${\text {CR}}$-functions defined on certain subsets of the boundary of a domain in ${{\mathbf {C}}^n}$.