Endpoint Estimates of Generalized Homogeneous Littlewood–Paley g-functions over Non-Homogeneous Metric Measure Spaces

Let(X,d,μ) 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 gr(r∈[2,∞)) is bounded from Hardy space H~1(μ) into L~1(μ).Moreover,the authors show that,if f∈RBMO(μ),then [gr(f)]~r is either infinite everywhere or finite almost everywhere,and in the latter case,[gr(f)]~r belongs to RBLO(μ) with the norm no more than ||f||_(RBMO(μ)) multiplied by a positive constant which is independent of f.As a corollary,the authors obtain the boundedness of gr from RBMO(μ) into RBLO(μ).The vector valued Calderón–Zygmund theory over(X,d,μ) is also established with details in this paper.