Strongly singular integral operators on the Heisenberg group: L2-estimates

We consider the L2 mapping properties of strongly singular integral operators on the Heisenberg group H n; these are convolution operators on H n whose kernels are too singular at the origin to be of Calderon and Zygmund type. This strong singularity in compensated for by introducing a suitably large oscillation. The analogous results in Rd are classical; these operators were first studied in the case d = 1 by Hirschman, then extended to higher dimensions by Wainger, and finally completed by Fefferman and Stein. Their results rely on the Fourier transform and asymptotic expressions for Bessel functions. In an analogous manner the results in the Heisenberg group setting are obtained by utilizing the group Fourier transform and uniform asymptotic forms for Laguerre functions due to Erdelyi.