A Dichotomy Concerning Uniform Boundedness of Riesz Transforms on Riemannian Manifolds

Journal article (2019)

Authors

Alex Amenta Analysis - EEMCS
Leonardo Tolomeo The University of Edinburgh
Research Group
Analysis (EEMCS) (TU Delft)
Copyright: © 2019 Alex Amenta, Leonardo Tolomeo
DOI
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https://doi.org/10.1090/proc/14730
Brownian motion Riesz transform Riemannian manifolds
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Published Date

2019

Language

English

Reuse Rights

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Analysis

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.