IUMJ

Title: $L^p$-estimates for Riesz transforms on forms in the Poincare space

Authors: Joaquim Bruna

Issue: Volume 54 (2005), Issue 1, 153-186

Abstract:

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 $m\neq(n\pm1)/2,n/2$.