  In [24, Theorem 2'] Charles Fefferman has proved the weak (1,1) boundedness
for a class of oscillating singular integrals that includes the oscillating
spectral multipliers of the Euclidean Laplacian $\Delta,$ namely, operators of
the 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 of
this work is to extend Fefferman's result to oscillating singular integrals on
any arbitrary compact Lie group. Applications to oscillating spectral
multipliers of the Laplace-Beltrami operator are also considered. The proof of
our main theorem illustrates the delicate relationship between the condition on
the kernel of the operator, its Fourier transform (defined in terms of the
representation theory of the group) and the microlocal/geometric properties of
the group.
