Published online by Cambridge University Press: 22 January 2016
1.In the study on Hölder-continuity of Brownian motion, A.N.Kol-mogorov introduced the concept of upper and lower classes and presented a criterion with the integral form to test whether some function belongs to upper or lower class; the so-called Kolmogorov’s test (I.Petrovesky [10]). P.Lévy considered the upper and lower class with regard to the uniform continuity of Brownian motion. We shall recall the definition of the upper and lower classes. We shall call <p(t) a function belonging to the upper class with regard to the uniform continuity of Brownian motion x(t) if there exists a positive number s{w) such that, for almost all w,
implies
(1.1)