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


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Abstract

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 \(\hat{f}(\xi)\) 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; ℚp), 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.

About the authors

Sergey S. Platonov

Institute of Mathematics

Author for correspondence.
Email: ssplatonov@yandex.ru
Russian Federation, Lenina av., 33, Petrozavodsk, 185910

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