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Riesz Transforms on Groups of Heisenberg Type

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Abstract

The aim of this article is to prove the following theorem.

Theorem Let p be in (1,∞), ℍ n,m a group of Heisenberg type, ℛ the vector of the Riesz transforms on ℍ n,m . There exists a constant C p independent of n and m such that for every f∈L p(ℍ n,m )

$$C_p^{-1}e^{-0.45m}\|f\|_{L^p(\mathbb{H}_{n,m})}\leq\||\mathcal{R}f|\|_{L^p(\mathbb{H}_{n,m})}\leq C_pe^{0.45m}\|f\|_{L^p(\mathbb{H}_{n,m})}.$$

It has been proved by F. Lust-Piquard that the operator norm of ℛ does not depend on n (see Lust-Piquard in Publ. Mat. 48(2):309–333, 2004), however its behavior in m has been entirely unknown.

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Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Helena Barbas

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  1. Helena Barbas
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Correspondence to Helena Barbas.

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Communicated by Marco Peloso.

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Barbas, H. Riesz Transforms on Groups of Heisenberg Type. J Geom Anal 20, 1–38 (2010). https://doi.org/10.1007/s12220-009-9097-4

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  • DOI: https://doi.org/10.1007/s12220-009-9097-4

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