Fourier Transform of Dini-Lipschitz Functions on Locally Compact Vilenkin Groups

Let $$G$$ be a locally compact bounded Vilenkin group, $$\Gamma$$ be the dual group of $$G$$ . Suppose that a function $$f(x)$$ belongs to the the Lebesgue class $$L^p(G)$$ , $$10$$ , $$\beta\in{\mathbb R}$$ , then for which values of $$r$$ we can guarantee that $$\widehat{f}\in L^r(\Gamma)$$ ? The result is an analogue of one classical theorem of E. Titchmarsh about the Fourier transform of functions from the Lipschitz classes on $${\mathbb R}$$ .