Strongly singular integral operators associated to different quasi-norms on the Heisenberg group

Abstract

In this article we study the behavior of strongly singular integrals associated to three different, albeit equivalent, quasi-norms on Heisenberg groups; these quasi-norms give rise to phase functions whose mixed Hessians may or may not drop rank along suitable varieties. In the particular case of the Koranyi norm we improve on previous arguments of the second author and obtain sharp $L^2$ estimates for the associated operators.