GENERALIZED FRACTIONAL INTEGRALS AND THEIR COMMUTATORS OVER NON-HOMOGENEOUS METRIC MEASURE SPACES

• Published in: Taiwanese journal of mathematics Vol. 18; no. 2; pp. 509 - 557
• Main Authors: Fu, Xing, Yang, Dachun, Yuan, Wen
• Format: Journal Article
• Language: English
• Published: Mathematical Society of the Republic of China 01.04.2014
• Subjects: Commutators; Euclidean space; Grants; Harmonic analysis; Interpolation; Lebesgue spaces; Mathematical integrals; Mathematical theorems; Orlicz space; Polynomials
• ISSN: 1027-5487; 2224-6851
• DOI: 10.11650/tjm.18.2014.3651

Let (χ, d, μ) be a metric measure space satisfying both the upper doubling and the geometrically doubling conditions. In this paper, the authors establish some equivalent characterizations for the boundedness of fractional integrals over (χ, d, μ). The authors also prove that multilinear commutators of fractional integrals with ΚΒΜO(μ) functions are bounded on Orlicz spaces over (χ, d, μ), which include Lebesgue spaces as special cases. The weak type endpoint estimates for multilinear commutators of fractional integrals with functions in the Orlicz-type space Osc exp   L r ( μ ) , wherer∈ [1, ∞), are also presented. Finally, all these results are applied to a specific example of fractional integrals over non-homogeneous metric measure spaces. 2010Mathematics Subject Classification: Primary 47B06; Secondary 47B47, 42B25, 42B35, 30L99. Key words and phrases: Non-homogeneous metric measure space, Fractional integral, Commutator, Orlicz space, Hardy space, RBMO(μ), Osc exp   L r ( μ ) .