$L^p$-estimates for Riesz transforms on forms in the Poincar\'e space

Abstract

Using hyperbolic form convolution with doubly isometry-invariant kernels, the explicit expression of the inverse of the de Rham laplacian A acting on m-forms in the Poincare space H n is found. Also, by means of some estimates for hyperbolic singular integrals, L p -estimates for the Riesz transforms ⊇ i Δ -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 H s m,p (H n ) in case m ¬= (n ± 1)/2, n/2.