  In this note we extend to metrizable profinite groups the classical theorems
of Titchmarsh on the Fourier transform of H\"older-Lipschitz functions. This
generalizes the results of Younis on compact zero-dimensional abelian groups to
the noncommutative case, and proves a relation between the
H\"older-Lipschitz-continuity of functions and their Sobolev regularity given
in terms of the Vladimirov-Taibleson operator. Since the class of profinite
groups is fairly big, the formulation of our results requires to impose a
special condition on the representation theory of the group. We prove that in
particular such condition is satisfied by compact nilpotent metrizable
profinite groups, which covers the case of compact nilpotent $\ell$-adic Lie
groups. In addition, we study the modulus of continuity of $L^2$-functions on
the group, the functional spaces related to it, and its relation to the
$L^2$-based H\"older-Lipschitz spaces. Finally, we also derive a
characterization for Dini-Lipschitz classes on metrizable profinite groups in
terms of the behavior of their Fourier coefficients.
