  Let $U$ be a unitary operator defined on some infinite-dimensional complex
Hilbert space ${\cal H}$. Under some suitable regularity assumptions, it is
known that a local positive commutation relation between $U$ and an auxiliary
self-adjoint operator $A$ defined on ${\cal H}$ allows to prove that the
spectrum of $U$ has no singular continuous spectrum and a finite point
spectrum, at least locally. We prove that under stronger regularity hypotheses,
the local regularity properties of the spectral measure of $U$ are improved,
leading to a better control of the decay of the correlation functions. As shown
in the applications, these results may be applied to the study of periodic
time-dependent quantum systems, classical dynamical systems and spectral
problems related to the theory of orthogonal polynomials on the unit circle.
