On the complex Abel-Radon transform for a given multidegree in the projective space

For a given multidegree d=(d_1,...,d_k), there is a natural affine covering
of the variety G of reduced d-complete intersections in P^N. Let X in P^N a
given projective subvariety of dimension n. For any domain U on G, we
associate a dual domain U^* on X, union of complete intersections of
dimension p=n-k on X.
If the current T is given on U^*, there is a naturally defined Abel-Radon
transform R(T), which is a holomorphic form of degree q, if T is d'-closed
of bidegree (q+p,p).
We prove the following result for the bidegree (n,p), when T is a locally
residual current:
Theorem. If R(T) extends meromorphically (resp. holomorphically) to a
greater domain U' on G, then T extends to (U')^* as a locally residual
current (resp. a d'-closed locally residual current).
After recalling the definition of locally residual currents, we show that
any
cohomology class H^i(U^*, w^q) can be for any q represented by a locally
residual current
for i<p, but this is in general no more the case for i>=p.