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$% O. Blasco and M. A. Pérez, On functions of integrable mean oscillation, Rev. Mat. Complut. 18 (2005), no. 2, 465–477.%$

On Functions of Integrable Mean Oscillation

Oscar BLASCO and M. Amparo PÉREZ
Departamento de Análisis Matemático
Universidad de Valencia
46100 Burjassot
Valencia Spain

Received: December 1, 2004
Accepted: April 21, 2005

Given f  (- L1(T)  we denote by wmo(f)  the modulus of mean oscillation given by

wmo(f)(t)=  sup -1  | f(eih)- mI(f)| dh
          0<|I| <t|I| I            2p

where I  is an arc of T  , |I| stands for the normalized length of I  , and         1- integral   ih dh-
mI(f) = |I| If(e )2p  . Similarly we denote by who(f)  the modulus of harmonic oscillation given by

                  integral   ih             ih dh
who(f)(t) =1-tsu<p|z|<1 T| f(e )- P(f)(z)|Pz(e )2p

where Pz(eih)  and P(f)  stand for the Poisson kernel and the Poisson integral of f  respectively.

It is shown that, for each 0< p < oo  , there exists Cp > 0  such that

 integral  1              integral  1               integral  1
   [wmo(f)(t)]pdt<   [who(f)(t)]pdt-< Cp   [wmo(f)(t)]pdt.
 0           t    0          t      0           t

Key words: mean oscillation, BMO, modulus of continuity.
2000 Mathematics Subject Classification: