$L^p$-estimates for Riesz transforms on forms in the Poincare space Joaquim Bruna 53C2158J0558J5058J70Hodge-de Rham laplacianSobolev spacesRiesz transformshyperbolic for convolution Using hyperbolic form convolution with doubly isometry-invariant kernels, the explicit expression of the inverse of the de Rham laplacian $\Delta$ acting on $m$-forms in the Poincar\'e space $\Hn$ is found. Also, by means of some estimates for hyperbolic singular integrals, $L^p$-estimates for the Riesz transforms $\nabla^i\Delta^{-1}$, $i\leq2$, in a range of $p$ depending on $m,n$ are obtained. Finally, using these, it is shown that $\Delta$ defines topological isomorphisms in a scale of Sobolev spaces $H^s_{m,p}(\Hn)$ in case $m\neq(n\pm1)/2,n/2$. Indiana University Mathematics Journal 2005 text pdf 10.1512/iumj.2005.54.2501 10.1512/iumj.2005.54.2501 en Indiana Univ. Math. J. 54 (2005) 153 - 186 state-of-the-art mathematics http://iumj.org/access/