Homogeneous spaces adapted to singular integral operators involving rotations

Calder\'on-Zygmund decompositions of functions have been used to prove weak-type (1,1) boundedness of singular integral operators. In many examples, the decomposition is done with respect to a family of balls that corresponds to some family of dilations. We study singular integral operators $T$ that require more particular families of balls, providing new spaces of homogeneous type. Rotations play a decisive role in the construction of these balls. Boundedness of $T$ can then be shown via Calder\'on-Zygmund decompositions with respect to this space of homogeneous type. We prove weak-type (1,1) and $\LP^p$ estimates for operators $T$ acting on $\LP^p(G)$, where $G$ is a homogeneous Lie group. Our results apply to the setting where the underlying group is the Heisenberg group and the rotations are symplectic automorphisms. They also apply to operators that arise from some hydrodynamical problem involving rotations.