Potential Analysis

, Volume 46, Issue 3, pp 463–484 | Cite as

Weak Type (1,1) Bounds for Some Operators Related to the Laplacian with Drift on Real Hyperbolic Spaces

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Abstract

The setting of this work is the n-dimensional hyperbolic space \(\mathbb {R}^{+} \times \mathbb {R}^{n-1}\), where the Laplacian is given a drift in the \(\mathbb {R}^{+}\) direction. We consider the operators defined by the horizontal Littlewood-Paley-Stein functions for the heat semigroup and the Poisson semigroup, and also the Riesz transforms of order 1 and 2. These operators are known to be bounded on \(L^{p},\; 1<p<\infty \), for the relevant measure. We show that most of the Littlewood-Paley-Stein operators and all the Riesz transforms are also of weak type (1,1). But in some exceptional cases, we disprove the weak type (1,1).

Keywords

Littlewood-Paley-Stein function Riesz transform Laplacian with drift Real hyperbolic space 

Mathematics Subject Classification (2010)

Primary 42B25 Secondary 58J35 

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© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.School of Mathematical SciencesFudan UniversityShanghaiChina
  2. 2.Mathematical SciencesUniversity of GothenburgGöteborgSweden
  3. 3.Mathematical SciencesChalmers University of TechnologyGöteborgSweden

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