Cauchy-Riemann equations on some lunes in

Let Ω = D\D1 where D and D1 are smoothly bounded strictly pseudoconvex domains in ; we suppose that the boundaries ∂D and ∂D 1 intersect each other transversally in the real sense and that ∂D⋂∂D 1 is contained in the zero set H of the real part of a function holomorphic in a neighbourhood of ΩH intersecting ∂D and ∂D 1 transversally. We give a homotopy formula for , for (p q)-forms on Ω(0≤p,q≤n). If f is a -closed (p,q)-form on Ω, with 1≤q≤n−3, we get from the homotopy formula a solution u of the equation is given by an integral operator T. An estimate is proved for Tf when the data f has bounded coefficients on Ω. Short abstract: For suitable bidegrees, we solve, by means of integral formulas, the -equation in non pseudoconvex domains which are differences of two strictly pseudo-convex domains and satisfy some natural geometric constraint. Estimates are given for the solution when the data has bounded coefficients. on