Titchmarsh Theorems for H\"older-Lipschitz functions on profinite groups

Abstract

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.