## 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 $L^p$.## References

- A. Amenta,
*New Riemannian manifolds with $L^p$-unbounded Riesz transform for $p>2$*, arXiv:1707.09781, July 2017. - Pascal Auscher and Thierry Coulhon,
*Riesz transform on manifolds and Poincaré inequalities*, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)**4**(2005), no. 3, 531–555. MR**2185868** - Pascal Auscher, Thierry Coulhon, Xuan Thinh Duong, and Steve Hofmann,
*Riesz transform on manifolds and heat kernel regularity*, Ann. Sci. École Norm. Sup. (4)**37**(2004), no. 6, 911–957 (English, with English and French summaries). MR**2119242**, DOI 10.1016/j.ansens.2004.10.003 - Frédéric Bernicot and Dorothee Frey,
*Riesz transforms through reverse Hölder and Poincaré inequalities*, Math. Z.**284**(2016), no. 3-4, 791–826. MR**3563255**, DOI 10.1007/s00209-016-1674-1 - Gilles Carron,
*Riesz transform on manifolds with quadratic curvature decay*, Rev. Mat. Iberoam.**33**(2017), no. 3, 749–788. MR**3713030**, DOI 10.4171/RMI/954 - Gilles Carron, Thierry Coulhon, and Andrew Hassell,
*Riesz transform and $L^p$-cohomology for manifolds with Euclidean ends*, Duke Math. J.**133**(2006), no. 1, 59–93. MR**2219270**, DOI 10.1215/S0012-7094-06-13313-6 - Li Chen, Thierry Coulhon, Joseph Feneuil, and Emmanuel Russ,
*Riesz transform for $1\le p\le 2$ without Gaussian heat kernel bound*, J. Geom. Anal.**27**(2017), no. 2, 1489–1514. MR**3625161**, DOI 10.1007/s12220-016-9728-5 - Thierry Coulhon and Xuan Thinh Duong,
*Riesz transforms for $1\leq p\leq 2$*, Trans. Amer. Math. Soc.**351**(1999), no. 3, 1151–1169. MR**1458299**, DOI 10.1090/S0002-9947-99-02090-5 - Thierry Coulhon and Xuan Thinh Duong,
*Riesz transform and related inequalities on noncompact Riemannian manifolds*, Comm. Pure Appl. Math.**56**(2003), no. 12, 1728–1751. MR**2001444**, DOI 10.1002/cpa.3040 - K. Dahmani, K. Domelevo, and S. Petermichl,
*Dimensionless ${L^p}$ estimates for the Riesz vector on manifolds*, arXiv:1802.00366, February 2018. - Baptiste Devyver,
*A perturbation result for the Riesz transform*, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)**14**(2015), no. 3, 937–964. MR**3445205** - Loukas Grafakos,
*Classical Fourier analysis*, 3rd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2014. MR**3243734**, DOI 10.1007/978-1-4939-1194-3 - Hong-Quan Li,
*La transformation de Riesz sur les variétés coniques*, J. Funct. Anal.**168**(1999), no. 1, 145–238 (French). MR**1717835**, DOI 10.1006/jfan.1999.3464 - Hong-Quan Li and Jie-Xiang Zhu,
*A note on “Riesz transform for $1 \le p \le 2$ without Gaussian heat kernel bound”*, J. Geom. Anal.**28**(2018), no. 2, 1597–1609. MR**3790512**, DOI 10.1007/s12220-017-9879-z - S. K. Pichorides,
*On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov*, Studia Math.**44**(1972), 165–179. (errata insert). MR**312140**, DOI 10.4064/sm-44-2-165-179 - E. M. Stein,
*Some results in harmonic analysis in $\textbf {R}^{n}$, for $n\rightarrow \infty$*, Bull. Amer. Math. Soc. (N.S.)**9**(1983), no. 1, 71–73. MR**699317**, DOI 10.1090/S0273-0979-1983-15157-1 - Shing Tung Yau,
*Some function-theoretic properties of complete Riemannian manifold and their applications to geometry*, Indiana Univ. Math. J.**25**(1976), no. 7, 659–670. MR**417452**, DOI 10.1512/iumj.1976.25.25051

## 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