Weak type estimates for a singular convolution operator on the Heisenberg group

Abstract:

On the Heisenberg group ${\mathbb {H}^n}$ with coordinates $(z,t) \in {\mathbb {C}^n} \times \mathbb {R}$, define the distribution $K(z,t) = L(z)\delta (t)$, where $L(z)$ is a homogeneous distribution on ${\mathbb {C}^n}$ of degree $- 2n$ , smooth away from the origin and $\delta (t)$ is the Dirac mass in the $t$ variable. We prove that the operator given by convolution with $K$ maps ${H^1}({\mathbb {H}^n})$ to weak ${L^1}({\mathbb {H}^n})$.