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In this paper we extend classical Titchmarsh theorems on the Fourier transform of Hölder–Lipschitz functions to the setting of compact homogeneous manifolds. As an application, we derive a Fourier multiplier theorem for $$L^2$$-Hölder–Lipschitz spaces on compact Lie groups. We also derive conditions and a characterisation for Dini–Lipschitz classes on compact homogeneous manifolds in terms of the behaviour of their Fourier coefficients.

Titchmarsh theorems for Fourier transforms of Hölder–Lipschitz functions on compact homogeneous manifolds