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Necessary condition for the L2 boundedness of the Riesz transform on Heisenberg groups

Part of: Classical measure theory Harmonic analysis in several variables

Published online by Cambridge University Press:  23 May 2023

DAMIAN DĄBROWSKI  and
MICHELE VILLA
DAMIAN DĄBROWSKI
Affiliation:
University of Jyväskylä, P.O. Box 35 (MaD), 40014, Finland e-mail: damian.m.dabrowski@jyu.fi
MICHELE VILLA
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C Facultat de Ciències, 08193 Bellaterra, Barcelona, Catalonia, Spain and Mathematics Research Unit, University of Oulu. P.O.Box 8000 FI 90014, U.S.A. e-mail: michele.villa@oulu.fi
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Abstract

Let $\mu$ be a Radon measure on the nth Heisenberg group ${\mathbb{H}}^n$. In this note we prove that if the $(2n+1)$-dimensional (Heisenberg) Riesz transform on ${\mathbb{H}}^n$ is $L^2(\mu)$-bounded, and if $\mu(F)=0$ for all Borel sets with ${\text{dim}}_H(F)\leq 2$, then $\mu$ must have $(2n+1)$-polynomial growth. This is the Heisenberg counterpart of a result of Guy David from [Dav91].

MSC classification

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 28A75: Length, area, volume, other geometric measure theory
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 175 , Issue 2 , September 2023 , pp. 445 - 458
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

†

Supported by Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R&D (grant MDM-2014-0445). Partially supported by the Catalan Agency for Management of University and Research Grants (grant 2017-SGR-0395), and by the Spanish Ministry of Science, Innovation and Universities (grant MTM-2016-77635-P).

‡

Supported 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.

§

Partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.

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