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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A dichotomy concerning uniform boundedness of Riesz transforms on Riemannian manifolds
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by Alex Amenta and Leonardo Tolomeo PDF
Proc. Amer. Math. Soc. 147 (2019), 4797-4803 Request permission

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
  • MR Author ID: 1089937
  • 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
  • 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
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4797-4803
  • MSC (2010): Primary 42B20; Secondary 58J35, 58J65
  • DOI: https://doi.org/10.1090/proc/14730
  • MathSciNet review: 4011513