The L2-order of magnitude of Vilenkin-Fourier coefficients

Let G be a compact, metrizable, zero-dimensional, abelian gruop,  i.e ., a Vilenkin group. It is well known ([2], [6] for example) that if f belongs to the Lipschitz class Lip (∝, p, G), 0 ≤ ∝ ≤  1, 1 < p ≤ 2, then its Fourier transform f  belongs to ℓ^B (Ĝ) for p/(p + ∝p - 1) < β ≤ p´ = p/(p - 1), where Ĝ is the dual of G. For Lipschitz functions on the real line ℝ and on the circle group T ([3], Theorem 85, p. 117; [5], Theorem (1.3) c, p. 108), the special case p = 2,  0 < ∝ <1, reveals some reversibility between the conditions on f and f. In the present work we extend, among other things, this reversibility to the L² Lipschitz functions on Vilenkin groups.