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

Abstract

Let ℚ_p be the field of p-adic numbers, a function f(x) belongs to the the Lebesgue class L^ρ(ℚ_p), 1 ρ ≤ 2, and let \(\hat{f}(\xi)\) 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; ℚ_p), α > 0, β ∈ ℝ, then for which values of r we can guarantee that \(\hat{f} \in {L^r}(\mathbb{Q}_p)\)? The result is an analogue of one classical theorem of E. Titchmarsh about the Fourier transform of functions from the Lipschitz classes on ℝ.