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

For a Young function I similar to with 0 a parts per thousand currency sign alpha < 1, let M (alpha,I similar to) be the fractional Orlicz maximal operator defined in the context of the spaces of homogeneous type (X, d, A mu) by M (alpha,I similar to) f(x) = sup (xaB) A mu(B) (alpha) aEuro-faEuro-(I similar to,B) , where aEuro-faEuro-(I similar to,B) is the mean Luxemburg norm of f on a ball B. When alpha = 0 we simply denote it by M (I similar to). In this paper we prove that if I broken vertical bar and I are two Young functions, there exists a third Young function I similar to such that the composition M (alpha,I) a similar to M (I broken vertical bar) is pointwise equivalent to M (alpha,I similar to). As a consequence we prove that for some Young functions I similar to, if M (alpha,I similar to) f < a a.e. and delta a (0, 1) then (M (alpha,I similar to) f) (delta) is an A (1)-weight.