Riesz transform on manifolds and Poincar\'{e} inequalities
  We study the validity of the $L^p$ inequality for the Riesz transform when $p>2$ and of its reverse inequality when $p<2$ on complete Riemannian manifolds under the doubling property and some Poincar\'e inequalities. 
Pascal Auscher (LM-Orsay), Thierry Coulhon (AGM)