The long open question regarding the almost everywhere convergence of the Fourier series of an L2-function having been recently settled by Carleson (1), an interesting problem still awaits solution. It is to investigate whether the Fourier series of a function of class Lp (1 < p < 2), converges (or diverges) almost everywhere; for it is already known that the Fourier series of an integrable function may diverge almost everywhere. See Kolmogorov (3) and Marcinkiewicz (4) or ((8), 1, p. 305), see also Kahane(2). The object of this paper is simply to show that a suitable restriction imposed on the modulus of smoothness of an integrable function guarantees the almost everywhere convergence of its Fourier series. This will establish, in a more general form, the author's conjecture made in (5). Our proof is based on Carleson's theorem according to which the Fourier series of an L2-function converges almost everywhere.