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.

中文翻译：

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p-Adic 数域上 Dini-Lipschitz 函数的傅里叶变换

设ℚp为p进数域，函数f(x)属于勒贝格类Lρ(ℚp)，1ρ≤2，令$$\hat{f}(\xi)$$为f 的傅立叶变换。在本文中，我们给出了下一个问题的答案：如果函数 f 属于 Dini-Lipschitz 类 DLip(α, β, ρ; ℚp), α > 0, β ∈ ℝ, 那么对于 r 的哪些值我们可以保证 $$\hat{f} \in {L^r}(\mathbb{Q}_p)$$? 结果是 E. Titchmarsh 关于 ℝ 上 Lipschitz 类函数的傅立叶变换的一个经典定理的类比。