Homotopy formulas for the $\overline\partial$-operator on $\mathbf CP^n$ and the Radon–Penrose transform

Global integral representations are constructed for differential forms on domains in complex projective space $\mathbf CP^n$.
Consequences of these representations are the following: first, criteria for the solvability of the inhomogeneous Cauchy–Riemann equations on $q$-pseudoconvex and $q$-pseudoconcave domains in an algebraic manifold; second, explicit formulas and bounds for solutions of these equations; and third, a description of the kernel and image and an inversion formula for the Radon-Penrose transform of $(0,q)$-forms on $q$-linearly concave domains in $\mathbf CP^n$.
Bibliography: 23 titles.