Strongly Singular Convolution Operators on the Heisenberg Group

We consider the L mapping properties of a model class of strongly singular integral operators on the Heisenberg group H; these are convolution operators on H 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 Rd These are convolution operators whose kernels are too singular at the origin to be of CalderónZygmund 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 be a distribution on Rd that away from the origin agrees with the function K(x) = |x|−d−αei|x|χ(|x|), where β > 0 and χ is smooth cut off function which equals one near the origin. Operators of this type were first studied by Hirschman [7] in the case d = 1 and then in higher dimensions by Wainger [12], Fefferman [3], and Fefferman and Stein [4]. 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, Ŝf(ξ) = f̂(ξ) ·m(ξ), where ̂ denotes the Fourier transform and m = K̂, the fact that K is a compactly supported distribution ensures that m(ξ) is a function. Plancherel’s theorem then implies that ‖Sf‖L2(Rd) ≤ A‖f‖L2(Rd) if and only if |m(ξ)| ≤ A, uniformly in ξ. Since K is also radial, i.e. K(x) = K0(x) for some function K0, we have m(ξ) = (2π) d 2 ∫ ∞ 0 K0(r)J d−2 2 (r|ξ|)(r|ξ|) 2−d 2 rd−1dr, 1 The distribution-valued function α 7→ K, initially defined for Reα < 0, continues analytically to all of C. 1