Spaces of type BLO on non-homogeneous metric measure

Abstract

Let ([inline-graphic not available: see fulltext], d, µ) be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. In this paper, we introduce the space RBLO(µ) and prove that it is a subset of the known space RBMO(µ) in this context. Moreover, we establish several useful characterizations for the space RBLO(µ). As an application, we obtain the boundedness of the maximal Calderón-Zygmund operators from L ^∞(µ) to RBLO(µ).

Keywords

Upper doubling / geometrically doubling / RBLO(µ) / maximal operator / Calderón-Zygmund maximal operator

References

[1.] Benett C. Another characterization of BLO. Proc Amer Math Soc, 1982, 85, 552-556

[2.] Coifman R. R., Rochberg R. Another characterization of BMO. Proc Amer Math Soc, 1980, 79, 249-254

[3.] Coifman R. R., Weiss G. Analyse Harmonique Non-commutative sur Certains Espaces Homogènes, 1971, Berlin: Springer.

[4.] Coifman R. R. Weiss G. Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc, 1977, 83, 569-645

[5.] Hajłasz P., Koskela P. Sobolev met Poincaré, 2000, Providence: Amer Math Soc.

[6.] Heinenon J. Lectures on Analysis on Metric Spaces, 2001, New York: Springer-Verlag.

[7.] Hu G., Yang D., Yang Dongyong. h ¹, bmo, blo and Littlewood-Paley g-functions with non-doubling measures. Rev Mat Ibero, 2009, 25, 595-667

[8.] Hytönen T. A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa. Publ Mat, 2010, 54, 485-504

[9.] Hytönen T, Liu Suile, Yang Dachun, Yang Dongyong. Boundedness of Calderón-Zygmund operators on non-homogeneous metric measure spaces. Canad J Math (to appear) or arXiv: 1011.2937

[10.] Hytönen T, Martikainen H. Non-homogeneous Tb theorem and random dyadic cubes on metric measure spaces. arXiv: 0911.4387

[11.] Hytönen T, Yang Dachun, Yang Dongyong. The Hardy space H ¹ on non-homogeneous metric spaces. arXiv: 1008.3831

[12.] Jiang Y. Spaces of type BLO for non-doubling measures. Proc Amer Math Soc, 2005, 133, 2101-2107

[13.] Leckband M. A. Structure results on the maximal Hilbert transform and two-weight norm inequalities. Indiana Univ Math J, 1985, 34, 259-275

[14.] Luukkainen J., Saksman E. Every complete doubling metric space carries a doubling measure. Proc Amer Math Soc, 1998, 126, 531-534

[15.] Nazarov F., Treil S., Volberg V. The Tb-theorem on non-homogeneous spaces. Acta Math, 2003, 190, 151-239

[16.] Stein E. M. Calderón A. P. Singular integral, harmonic functions, and differentiability properties of functions of several variables. Singular Integrals. Proc Symp Pure Math, 10, 1967, Providence: Amer Math Soc 316-335

[17.] Tolsa X. BMO, H ¹ and Calderón-Zygmund operators for non doubling measures. Math Ann, 2001, 319, 89-149

[18.] Tolsa X. Painlevé’s problem and the semiadditivity of analytic capacity. Acta Math, 2003, 190, 105-149

[19.] Tolsa X. Analytic capacity and Calderón-Zygmund theory with non doubling measures. Seminar of Mathematical Analysis, 2004, Seville: Univ Sevilla Secr Publ 239-271

[20.] Volberg A, Wick B D. Bergman-type singular operators and the characterization of Carleson measures for Besov-Sobolev spaces on the complex ball. Amer J Math (to appear) or arXiv: 0910.1142

[21.] Wu J. Hausdorff dimension and doubling measures on metric spaces. Proc Amer Math Soc, 1998, 126, 1453-1459