A Dichotomy Concerning Uniform Boundedness of Riesz Transforms on Riemannian Manifolds

Alex Amenta, Leonardo Tolomeo

Research output: Contribution to journalArticleScientificpeer-review

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Abstract

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

Original languageEnglish
Pages (from-to)4797-4803
Number of pages7
JournalAmerican Mathematical Society. Proceedings
Volume147
Issue number11
DOIs
Publication statusPublished - 2019

Keywords

  • Brownian motion
  • Riemannian manifolds
  • Riesz transform

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