Fourier Transform of Dini-Lipschitz Functions on Locally Compact Vilenkin Groups

摘要

Let G be a locally compact bounded Vilenkin group, Gamma be the dual group of G. Suppose that a function f(x) belongs to the the Lebesgue class L-p(G), 1 < p <= 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, p; G), 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(Gamma)? The result is an analogue of one classical theorem of E. Titchmarsh about the Fourier transform of functions from the Lipschitz classes on R.