Fourier Transform of Dini-Lipschitz Functions on the Field of p-Adic Numbers

Let ℚp be the field of p-adic numbers, a function f(x) belongs to the the Lebesgue class Lρ(ℚp), 1 ρ ≤ 2, and let f^(ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{f}(\xi)$$\end{document} be the Fourier transform of f. In this paper we give an answer to the next problem: if the function f belongs to the Dini-Lipschitz class DLip(α, β, ρ; ℚp), α > 0, β ∈ ℝ, then for which values of r we can guarantee that f^∈Lr(Qp)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{f} \in {L^r}(\mathbb{Q}_p)$$\end{document}? The result is an analogue of one classical theorem of E. Titchmarsh about the Fourier transform of functions from the Lipschitz classes on ℝ.