A dichotomy concerning uniform boundedness of Riesz transforms on Riemannian manifolds

TL;DR

This paper constructs a Riemannian manifold demonstrating a dichotomy in the boundedness of Riesz transforms, showing either uniform bounds exist across all manifolds of a given dimension or unboundedness can occur.

Contribution

The authors establish a dichotomy result for the uniform boundedness of Riesz transforms on Riemannian manifolds, providing a new understanding of their behavior across different geometries.

Findings

Existence of a manifold with unbounded Riesz transform on L^p

Dichotomy: either uniform boundedness or unboundedness for all manifolds of fixed dimension

Construction method for manifolds illustrating the dichotomy

Abstract

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