Let the manifold $X$ parametrise a family of compact complex submanifolds of the complex (or CR) manifold $Z$ . Under mild conditions the Penrose transform typically provides isomorphisms between a cohomology group of a holomorphic vector bundle $V\longrightarrow Z$ and the kernel of a differential operator between sections of vector bundles over $X$ . When the spaces in question are homogeneous for a group $G$ the Penrose transform provides an intertwining operator between representations.

The paper develops a Penrose transform for compactly supported cohomology on $Z$ . It provides a number of examples where a compactly supported cohomology group is shown to be isomorphic to the cokernel of a differential operator between compactly supported sections of vector bundles over $X$ . It considers also how the Serre duality pairing carries through the transform.

The Funk transform is the integral transform from the space of smooth even functions on the unit sphere S2⊂ℝ3 to itself defined by integration over great circles. One can regard this transform as a limit in a certain sense of the Penrose transform from [Copf ]ℙ2 to [Copf ]ℙ*ast;2. We exploit this viewpoint by developing a new proof of the bijectivity of the Funk transform which proceeds by considering the cohomology of a certain involutive (or formally integrable) structure on an intermediate space. This is the simplest example of what we hope will prove to be a general method of obtaining results in real integral geometry by means of complex holomorphic methods derived from the Penrose transform.