A dichotomy concerning uniform boundedness of Riesz transforms on Riemannian manifolds

Abstract

Given a sequence of complete Riemannian manifolds ( M n ) (M_n) of the same dimension, we construct a complete Riemannian manifold M M such that for all p ∈ ( 1 , ∞ ) p \in (1,\infty ) the L p L^p -norm of the Riesz transform on M M dominates the L p L^p -norm of the Riesz transform on M n M_n for all n n . Thus we establish the following dichotomy: given p p and d d , either there is a uniform L p L^p bound on the Riesz transform over all complete d d -dimensional Riemannian manifolds, or there exists a complete Riemannian manifold with Riesz transform unbounded on  L p L^p .