MR2524292 (2010f:26007): Kolyada, V. I.; Lind, M. On functions of bounded p-variation. J. Math. Anal. Appl. 356 (2009), no. 2, 582–604. (Reviewer: Luisa Di Piazza),

For p∈(1,+∞), let f∈Lp be a 1-periodic function on the real line, with the norm of f given by ∥f∥p=(∫10|f(x)|pdx)1/p. The Lp-modulus of continuity of f is defined by ω(f,δ)p=sup0≤h≤δ(∫10|f(x+h)−f(x)|pdx)1/p, 0≤δ≤1. A partition of period 1 (or simply a partition) is a set Π={x0,x1,…,xn} of points such that x0<x1<…<xn=x0+1. For a given partition Π={x0,x1,…,xn} let vp(f;Π)=(∑k=0n−1|f(xk+1)−f(xk)|p)1/p. The modulus of p-continuity of f is defined by ω1−1/p(f,δ)=sup∥Π∥≤δvp(f;Π), where the supremum is taken over all partitions Π such that ∥Π∥=maxk(xk+1−xk)≤δ. In this paper, improving a previous estimate given by A. P. Terehin [Mat. Zametki 2 (1967), 289--300; MR0223512 (36 #6560)], it is shown that ω1−1/p(f,δ)≤A[δ−1/pω(f,δ)p+1pp′∫δ0t−1/pω(f,δ)pdtt],(1) for any δ∈(0,1], where A is an absolute constant and, as usual, p′=p/(p−1). But the main result is the sharpness of the estimate (1). In fact an example is given of a function with an arbitrary prescribed order of the modulus of continuity in Lp, for which the opposite inequality holds for all δ∈[0,1]. Some other estimates involving the Lp-modulus of continuity are also proved. Reviewed by Luisa Di Piazza