An analogue of the Titchmarsh theorem for the Fourier transform on locally compact Vilenkin groups (Q1743665)

Suppose that \(f\in L^2(\mathbb{R})\), \(0<\alpha<1\). Titchmarsh's theorem states that \(f\) satisfies the \(L^2\) Hölder condition \[ \|f(\cdot - h) - f(\cdot)\|_{L^2(\mathbb{R})}=O(h^{\alpha}), \quad h\to 0, \] if and only if \[ \int_{|t|>r}|\widehat{f}(t)|^2\,dt = O(r^{-2\alpha}), \quad r\to\infty, \] where \(\widehat{f}\) is the Fourier transform of \(f\). The author proves an analogue of this theorem for functions on locally compact Vilenkin groups.