We construct a variant of Koppelman's formula for (0,q)-forms with values in a line bundle, O(l), on projective space. The formula is then applied to a study of a Radon transform for (0,q)-forms, introduced by Gindikin-Henkin-Polyakov. Our presentation follows along the basic lines of Henkin-Polyakov [3], with some simplifications.

In 1945 the first author introduced the classes $H^p\left(\alpha \right)$, 1 ≤ p<∞, α > -1, of holomorphic functions in the unit disk with finite integral (1) ∬ |f(ζ)|p (1-|ζ|²)α dξ dη < ∞ (ζ=ξ+iη) and established the following integral formula for $f\in H^p\left(\alpha \right)$: (2) f(z) = (α+1)/π ∬ f(ζ) ((1-|ζ|²)α)/((1-zζ̅)2+α) dξdη, z∈ . We have established that the analogues of the integral representation (2) hold for holomorphic functions in Ω from the classes $L^p(\Omega ;{\left[K\left(w\right)\right]}^{\alpha }dm\left(w\right))$, where: 1) $\Omega =w=(w₁,...,w_n)\in {ℂ}^n:Imw₁>{\sum }_{k=2}^n\left|w_k\right|²$, $K\left(w\right)=Imw₁-{\sum }_{k=2}^n\left|w_k\right|²$; 2) Ω is the matrix domain consisting of those complex m...