L^p-estimates for riesz transforms on forms in the poincaré space

Using hyperbolic form convolution with doubly isometry-invariant kernels, the explicit expression of the inverse of the de Rham laplacian Δ acting on m-forms in the Poincaré space ℍn is found. Also, by means of some estimates for hyperbolic singular integrals, Lp- estimates for the Riesz transforms ∇iΔ-1, i ≤ 2, in a range of p depending on m, n are obtained. Finally, using these, it is shown that A defines topological isomorphisms in a scale of Sobolev spaces Hm,ps(ℍn) in case m ≠ (n ± 1)/2, n/2.