On Functions of Integrable Mean Oscillation

Given f 2 L1(T) we denote by wmo(f) the modulus of mean oscillation given by wmo(f)(t) = sup 0<|I| t 1 |I| Z I |f(ei ) − mI (f)| d 2 where I is an arc of T, |I| stands for the normalized length of I, and mI (f) = 1 |I| R I f(ei ) d 2 . Similarly we denote by who(f) the modulus of harmonic oscillation given by who(f)(t) = sup 1−t |z|<1 Z T |f(ei ) − P(f)(z)|Pz(ei ) d 2 where Pz(ei ) and P(f) stand for the Poisson kernel and the Poisson integral of f respectively. It is shown that, for each 0 < p < 1, there exists Cp > 0 such that Z 1 0 [wmo(f)(t)]p dt t Z 1 0 [who(f)(t)]p dt t Cp Z 1 0 [wmo(f)(t)]p dt t.