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In [24, Theorem 2'] Charles Fefferman has proved the weak (1,1) boundednessfor a class of oscillating singular integrals that includes the oscillatingspectral multipliers of the Euclidean Laplacian $\Delta,$ namely, operators ofthe form \begin{equation}  T_{\theta}(-\Delta):= (1-\Delta)^{-\frac{n\theta}{4}}e^{i(1-\Delta)^{\frac{\theta}{2}}},\,0\leq \theta <1. \end{equation} The aim ofthis work is to extend Fefferman's result to oscillating singular integrals onany arbitrary compact Lie group. Applications to oscillating spectralmultipliers of the Laplace-Beltrami operator are also considered. The proof ofour main theorem illustrates the delicate relationship between the condition onthe kernel of the operator, its Fourier transform (defined in terms of therepresentation theory of the group) and the microlocal/geometric properties ofthe group.

Oscillating singular integral operators on compact Lie groups revisited