Abstract
AbstractIn the Heisenberg group of dimension $$2n+1$$
2
n
+
1
, we consider the sub-Laplacian with a drift in the horizontal coordinates. There is a related measure for which this operator is symmetric. The corresponding Riesz transforms are known to be $$L^p$$
L
p
bounded with respect to this measure. We prove that the Riesz transforms of order 1 are also of weak type (1, 1), and that this is false for order 3 and above. Further, we consider the related maximal Littlewood–Paley–Stein operators and prove the weak type (1, 1) for those of order 1 and disprove it for higher orders.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,General Mathematics,Analysis
Reference53 articles.
1. Agrachev, A., Barilari, D., Boscain, U.: A Comprehensive Introduction to Sub-riemannian Geometry, vol. 181. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2020)
2. Alexopoulos, G.: An application of homogenization theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth. Can. J. Math. 44, 691–727 (1992)
3. Alexopoulos, G.: Sub-laplacians with drift on lie groups of polynomial volume growth. Mem. Am. Math. Soc. 155(739), x+101 (2002)
4. Amenta, A., Tolomeo, L.: A dichotomy concerning uniform boundedness of Riesz transforms on Riemannian manifolds. Proc. Am. Math. Soc. 147, 4797–4803 (2019)
5. Anker, J.-P.: Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces. Duke Math. J. 65, 257–297 (1992)