Remarks on the multiplier operators associated with a cylindrical distance function

In this note, we consider L p and maximal L p estimates for the generalized Riesz means which are associated with the cylindrical distance function ρ ( ξ ) = max { | ξ ′ | , | ξ d + 1 | } , ( ξ ′ , ξ d + 1 ) ∈ R d × 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.