Composition of fractional Orlicz maximal operators and A 1-weights on spaces of homogeneous type

For a Young function Θ with 0 ≤ α < 1, let M _α,Θ be the fractional Orlicz maximal operator defined in the context of the spaces of homogeneous type (X, d, μ) by M _α,Θ f(x) = sup _x∈B μ(B) ^α ‖f‖_Θ,B , where ‖f‖_Θ,B is the mean Luxemburg norm of f on a ball B. When α = 0 we simply denote it by M _Θ. 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 M _α,Ψ ∘ M _Φ is pointwise equivalent to M _α,Θ. As a consequence we prove that for some Young functions Θ, if M _α,Θ f < ∞ a.e. and δ ∈ (0, 1) then (M _α,Θ f) ^δ is an A ₁-weight.