Kernels for the tangential cauchy-riemann equations

On certain codimension one and codimension two submanifolds in Cⁿ, we can solve the tangential Cauchy-Riemann equations (FORMULA PRESENTED) with an explicit integral formula for the solution. Let (FORMULA PRESENTED), where D is a strictly pseudoconvex domain in (FORMULA PRESENTED) M be defined by (FORMULA PRESENTED), where h is holomorphic near D. Points on the boundary of ω, ¶ω, where the tangent space of ¶ω becomes complex linear, are called characteristic points. Theorem 1. Suppose ¶ω is admissible (in particular if ¶ω has two characteristic points). Suppose (FORMULA PRESENTED), is smooth on ω and satisfies (FORMULA PRESENTED) on ω then there exists (FORMULA PRESENTED) which is smooth on ω except possibly at the characteristic points on ¶ω and which solves the equation (FORMULA PRESENTED) on ω. Theorem 2. Suppose (FORMULA PRESENTED), is smooth on ω; vanishes near each characteristic point, and (FORMULA PRESENTED) then there exists (FORMULA PRESENTED) satisfying (FORMULA PRESENTED). Theorem 3. Suppose (FORMULA PRESENTED), is smooth with compact support in and (FORMULA PRESENTED). Then there exists (FORMULA PRESENTED) compact support in ω and which solves (FORMULA PRESENTED). In all three theorems we have an explicit integral formula for the solution. Now suppose S = ¶ω. Let Cs be the set of characteristic points on S. We construct an explicit operator (FORMULA PRESENTED) with the following properties. Theorem 4. The operator E maps (FORMULA PRESENTED).