Remarks on the multiplier operators associated with a cylindrical distance function

In this note, we consider Lp and maximal Lp estimates for the generalized Riesz means which are associated with the cylindrical distance function ρ(ξ)=max{|ξ′|,|ξd+1|}, (ξ′,ξd+1)∈Rd×R. We prove these estimates up to the currently known range of the spherical Bochner-Riesz and its maximal operators. This is done by establishing implications between the corresponding estimates for the spherical Bochner-Riesz and the cylindrical multiplier operators.