Spherical means on the Heisenberg group: Stability of a maximal function estimate

Consider the surface measure $\mu$ on a sphere in a nonvertical hyperplane on the Heisenberg group ${\mathbb{H}}^n$, $n\ge 2$, and the convolution $f\ast \mu$. Form the associated maximal function $Mf={\sup }_{t>0}\mathrm{\mid }f\ast {\mu }_t\mathrm{\mid }$ generated by the automorphic dilations. We use decoupling inequalities due to Wolff and Bourgain-Demeter to prove $L^p$-boundedness of $M$ in an optimal range.