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

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).

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Correspondence to Peter Sjögren.

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Li, HQ., Sjögren, P. Weak Type (1,1) Bounds for Some Operators Related to the Laplacian with Drift on Real Hyperbolic Spaces. Potential Anal 46, 463–484 (2017). https://doi.org/10.1007/s11118-016-9590-x

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Keywords

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

Mathematics Subject Classification (2010)

  • Primary 42B25
  • Secondary 58J35