Weak type (1, 1) for some operators related to the Laplacian with drift on real hyperbolic spaces

Journal article, 2017

The setting of this work is the $n$-dimensional hyperbolic space $\R^+ \times \R^{n-1}$, where the Laplacian is given a drift in the $\R^+$ direction. We consider the operators given 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). In the exceptional cases, we disprove the weak type (1,1).