Endpoint estimates of generalized homogeneous Littlewood-Paley g-functions over non-homogeneous metric measure spaces

Let (X, d, .) be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions. Under the weak reverse doubling condition, the authors prove that the generalized homogeneous Littlewood.Paley g-function . r (r . [2,.)) is bounded from Hardy space H 1(.) into L 1(.). Moreover, the authors show that, if f . RBMO(.), then [. r (f)]r is either infinite everywhere or finite almost everywhere, and in the latter case, [. r (f)]r belongs to RBLO(.) with the norm no more than .f.RBMO(.) multiplied by a positive constant which is independent of f. As a corollary, the authors obtain the boundedness of . r from RBMO(.) into RBLO(.). The vector valued Calderón.Zygmund theory over (X, d, .) is also established with details in this paper.