On the integral moduli of continuity in 𝐿_{𝑝}(1<𝑝<∞) of Fourier series with monotone coefficients

Abstract

respectively, where II p denotes the norm in Lp. The Lipschitz and Zygmund classes Ap and A* are then defined by cop(h; f) = 0(h) and c4*(h; f) = 0(h) respectively. The problem of what can be said about the integral modulus of continuity (of first order) of the functions of the class A* (1 <p < oo) was solved by A. Timan and M. Timan [6] for p = 2 and in the general case by Zygmund [7] in the following way:'