Let (X, d,µ) be a metric measure space satisfying the upper doubling condition and the geometrically doubling condition in the sense of Hytönen. We prove that the Lp(µ)-boundedness with p ∈ (1,∞) of the Marcinkiewicz integral is equivalent to either of its boundedness from L1(µ) into L1,∞(µ) or from the atomic Hardy space H1(µ) into L1(µ). Moreover, we show that, if the Marcinkiewicz integral is bounded from H1(µ) into L1(µ), then it is also bounded from L∞(µ) into the space RBLO(µ) (the regularized BLO), which is a proper subset of RBMO(µ) (the regularized BMO) and, conversely, if the Marcinkiewicz integral is bounded from Lb∞ (µ) (the set of all L∞(µ) functions with bounded support) into the space RBMO(µ), then it is also bounded from the finite atomic Hardy space Hfin1,∞ (µ) into L1(µ). These results essentially improve the known results even for non-doubling measures.
upper doubling geometrically doubling Marcinkiewicz integral atomic Hardy space RBMO(µ)
42B20 42B25 42B35 30L99
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Fu X, Yang D, Yuan W. Boundedness of multilinear commutators of Calderón-Zygmund operators on Orlicz spaces over non-homogeneous spaces. Taiwanese J Math, 2012, 16: 2203–2238MATHMathSciNetGoogle Scholar
Lin H, Yang D. An interpolation theorem for sublinear operators on non-homogeneous metric measure spaces. Banach J Math Anal, 2012, 6: 168–179MATHMathSciNetGoogle Scholar
Liu S, Meng Y, Yang D. Boundedness of maximal Calderón-Zygmund operators on non-homogeneous metric measure spaces. Proc Roy Soc Edinburgh Sect A, 2014, in pressGoogle Scholar
Liu S, Yang D, Yang D. Boundedness of Calderón-Zygmund operators on non-homogeneous metric measure spaces: Equivalent characterizations. J Math Anal Appl, 2012, 386: 258–272CrossRefMATHMathSciNetGoogle Scholar