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

Abstract

For a Young function Θ with 0 ≤ α < 1, let Mα,Θ be the fractional Orlicz maximal operator defined in the context of the spaces of homogeneous type (X, d, μ) by Mα,Θf(x) = supx∈Bμ(B)α{double pipe}f{double pipe}Θ,B, where {double pipe}f{double pipe}Θ,B is the mean Luxemburg norm of f on a ball B. When α = 0 we simply denote it by MΘ. In this paper we prove that if Φ and Ψ are two Young functions, there exists a third Young function Θ such that the composition Mα,Ψ {ring operator} MΦ is pointwise equivalent to Mα,Θ. As a consequence we prove that for some Young functions Θ, if Mα,Θf <∞ a.e. and δ ∈ (0,1) then (Mα,Θf)δ is an A1-weight.