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

Let Q(p) be the field of p-adic numbers, a function f (x) belongs to the the Lebesgue class L-rho(Q(p)), 1 < rho <= 2, and let <((f)over cap>)(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(alpha, beta, rho; Q(p)), alpha > 0, beta is an element of R, then for which values of r we can guarantee that (f) over cap is an element of L-r (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.