Abstract. We consider the L 2 mapping properties of a model class 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 Calderón-Zygmund type. This strong singularity is compensated for by introducing a suitably large oscillation. Our results are obtained by utilizing the group Fourier transform and uniform asymptotic forms for Laguerre functions due to Erdélyi. Overture: Strongly singular convolution operators on R d These are convolution operators whose kernels are too singular at the origin to be of Calderón-Zygmund type. We choose to compensate for this strong singularity by introducing a suitably large oscillation. More precisely these are the operators given formally by Sf = f ∗ K, where K is a distribution on R d that away from the origin agrees with the function K(x) =|x | −d−α e i|x|−β χ(|x|), where β>0andχis smooth cut off function which equals one near the origin. 1 Operators of this type were first studied by Hirschman  in the case d =1andthen in higher dimensions by Wainger , Fefferman , and Fefferman and Stein . What is of interest here is the precise relationship between the size of the singularity and that of the required oscillation in order for S to extend to a bounded operator on L2 (Rd). Theorem A. S extends to a bounded operator on L2 (Rd) if and only if α ≤ dβ 2. Sketch of proof. Since S is translation invariant it may be realized as a Fourier multiplier, Sf(ξ) = f(ξ) · m(ξ), where denotes the Fourier transform and m = K,thefactthatKis a compactly supported distribution ensures that m(ξ) is a function. Plancherel’s theorem then implies that ‖Sf ‖ L 2 (R d) ≤ A‖f ‖ L 2 (R d) if and only if |m(ξ) | ≤A, uniformly in ξ
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