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A dichotomy concerning uniform boundedness of Riesz transforms on Riemannian manifolds


Authors: Alex Amenta and Leonardo Tolomeo
Journal: Proc. Amer. Math. Soc. 147 (2019), 4797-4803
MSC (2010): Primary 42B20; Secondary 58J35, 58J65
DOI: https://doi.org/10.1090/proc/14730
Published electronically: August 7, 2019
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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$.


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Additional Information

Alex Amenta
Affiliation: Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
Email: amenta@fastmail.fm

Leonardo Tolomeo
Affiliation: School of Mathematics, The University of Edinburgh and Maxwell Institute for Mathematical Sciences, James Clerk Maxwell Building, Rm 5210 The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom
Email: L.Tolomeo@sms.ed.ac.uk

DOI: https://doi.org/10.1090/proc/14730
Keywords: Riesz transform, Riemannian manifolds, Brownian motion
Received by editor(s): August 20, 2018
Received by editor(s) in revised form: January 24, 2019
Published electronically: August 7, 2019
Additional Notes: The first author was supported by the VIDI subsidy 639.032.427 of the Netherlands Organisation for Scientific Research (NWO). The second author was supported by the European Research Council (grant no. 637995 “ProbDynDispEq”) and by The Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University, and the University of Edinburgh.
Communicated by: Svitlana Mayboroda
Article copyright: © Copyright 2019 American Mathematical Society