Composition of Fractional Orlicz maximal operators and A1-weights on spaces of homogeneous type

Abstract

For a Young function $\Theta$ and $0\leq \alpha <1$, let $M_{\alpha, \Theta}$ be the fractional Orlicz maximal operator defined in the context of the spaces of homogeneous type $(X,d,\mu)$ by $M_{\alpha, \Theta}f(x) = \sup_{x\in B} \mu(B)^{\alpha}||f||_{\Theta,B}$, 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 $M_{\alpha, \Theta}f<\infty$ a.e. and $\delta \in (0,1)$ then $(M_{\alpha, \Theta}f)^{\delta}$ is an $A_1$-weight.

Accepted: Acta Mathematica Sinica. September 2009.