RMC - Vol. 18 Num. 2, Blasco/Pérez

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

On Functions of Integrable Mean Oscillation

Oscar BLASCO and M. Amparo PÉREZ

Departamento de Análisis Matemático
Universidad de Valencia
46100 Burjassot
Valencia — Spain

oblasco@uv.es

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

ABSTRACT

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

integral 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: 46B25.