Composition of fractional Orlicz maximal operators and A_1--weights on spaces of homogeneous type

A. BERNARDIS,M.LORENTE, G. PRADOLINI, M.S. RIVEROS,

ACTA MATHEMATICA SINICA-ENGLISH SERIES

SPRINGER HEIDELBERG

Año: 2010 vol. 26 p. 1509 - 1518

1439-8516

For a Young function $Theta$ and $alpha in [0,1)$, let $M_{alpha, Theta}$ be the fractional Orlicz maximal operator defined in the context of the spaces of homogeneous type $(X,d,mu)$ by $M_{alpha, Theta}f(x) = sup_{xin B} mu(B)^{alpha}||f||_{Theta,B}$, where $||f||_{Theta,B}$ is the mean Luxemburg norm of $f$ on a ball $B$. When $alpha=0$ we simply denote it by $M_{Theta}$. In this paper we prove that if $Phi$ and $Psi$ are two Young functions, there exists a third Young function $Theta$ such that the composition $M_{alpha,Psi}circ M_{Phi}$ is pointwise equivalent to $M_{alpha, Theta}$. As a consequence we prove that for some Young functions $Theta$, if $M_{alpha, Theta}f$ is finite a.e. and $delta in (0,1)$ then $(M_{alpha, Theta}f)^{delta}$ is an $A_1$-weight.