Com position of fractional Orlicz maximal operators and $A_1$-weights on spaces of homogeneous type

ANA BERNARDIS; MARÍA LORENTE; GLADIS PRADOLINI; MARÍA SILVINA RIVEROS

Acta Mathematica Sinica

Springer Berlin Heidelberg

Lugar: Beijing; Año: 2010 vol. 26 p. 1509 - 1518

1439-8516

For a Young function $Theta$ and $0leq alpha <1$, let$M_{alpha, Theta}$ be the fractional Orlicz maximal operatordefined 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 themean Luxemburg norm of $f$ on a ball $B$. When $alpha=0$ wesimply denote it by $M_{Theta}$. In this paper we prove that if$Phi$ and $Psi$ are two Young functions, there exists a thirdYoung 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 someYoung 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.