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An interpolation theorem for sublinear operators on non-homogeneous metric measure spaces. (English) Zbl 1252.42025
Let $(X,d,\mu)$ be a metric measure space with the geometric doubling property and the upper doubling condition for the measure $\mu$. In this setting, the regularized BMO space $\text{RBMO}(\mu)$ and the Hardy space $H^1(\mu)$ have been defined and studied in a number of recent papers. Here, the authors prove that any sublinear operator $T$ that is bounded from $H^1(\mu)$ to $L^{1,\infty}(\mu)$ and from $L^\infty(\mu)$ to $\text{RBMO}(\mu)$, is also bounded on $L^p(\mu)$ for all $p\in(1,\infty)$. This improves a result of {\it B. T. Anh} and {\it X. T. Duong} [“Hardy spaces, regularized BMO spaces and the boundedness of Calderón-Zygmund operators on non-homogeneous spaces”, \url{arXiv:1009.1274}, to appear in J. Geom. Anal.] who proved it for `linear’ instead of `sublinear’ and $L^1(\mu)$ instead of $L^{1,\infty}(\mu)$. The proof again uses the Calderón--Zygmund decomposition of Anh and Duong [op. cit.] in this setting, but also needs some new ideas.

42B35Function spaces arising in harmonic analysis
42B25Maximal functions, Littlewood-Paley theory
47B38Operators on function spaces (general)
Full Text: EMIS Euclid arXiv
[1] B.T. Anh and X.T. Duong, Hardy spaces, regularized BMO spaces and the boundeness of Calderón-Zygmund operators on non-homogeneous spaces , J. Geom. Anal., doi: 10.1007/s12220-011-9268-y. · Zbl 1267.42013
[2] R.R. Coifman and R. Rochberg, Another characterization of BMO , Proc. Amer. Math. Soc. 79 (1980), 249-254. · Zbl 0432.42016 · doi:10.2307/2043245
[3] R.R. Coifman and G. Weiss, Analyse Harmonique Non-commutative sur Certains Espaces Homogenes , Lecture Notes in Math. 242 , Springer, Berlin, 1971. · Zbl 0224.43006 · doi:10.1007/BFb0058946
[4] R.R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis , Bull. Amer. Math. Soc. 83 (1977), 569-645. · Zbl 0358.30023 · doi:10.1090/S0002-9904-1977-14325-5
[5] P. Hajłasz and P. Koskela, Sobolev met Poincaré , Mem. Amer. Math. Soc. 145 (2000), no. 688. · Zbl 0954.46022
[6] J. Heinenon, Lectures on Analysis on Metric Spaces , Springer-Verlag, New York, 2001. · Zbl 0985.46008 · doi:10.1007/978-1-4613-0131-8
[7] G. Hu, Y. Meng and D. Yang, A new characterization of regularized BMO spaces on non-homogeneous spaces and its applications , · Zbl 1278.42015
[8] T. Hytönen, A framework for non-homogeneous analysis on metric spaces, and the $\rbmo$ space of Tolsa , Publ. Mat. 54 (2010), 485-504. · Zbl 1246.30087 · doi:10.5565/PUBLMAT_54210_10
[9] T. Hytönen, S. Liu, Da. Yang and Do. Yang, Boundedness of Calderón-Zygmund operators on non-homogeneous metric measure spaces , Canad. J. Math. doi:10.4153/CJM-2011-065-2.
[10] T. Hytönen, Da. Yang and Do. Yang, The Hardy space $H^1$ on non-homogeneous metric spaces , Math. Proc. Cambridge Philos. Soc. 153 (2012), no. 1, 9-31. · Zbl 1250.42076 · doi:10.1017/S0305004111000776
[11] T. Hytönen and H. Martikainen, Non-homogeneous $Tb$ theorem and random dyadic cubes on metric measure spaces , J. Geom. Anal., doi: 10.1007/s1220-011-9230-z. · Zbl 1261.42017 · doi:10.1007/s12220-011-9230-z
[12] H. Lin and D. Yang, Spaces of type BLO on non-homogeneous metric measure spaces , Front. Math. China 6 (2011), 271-292. · Zbl 1217.42049 · doi:10.1007/s11464-011-0098-9
[13] S. Liu, Da. Yang and Do. Yang, Boundedness of Calderón-Zygmund operators on non-homogeneous metric measure spaces: equivalent characterizations , J. Math. Anal. Appl. 386 (2012), 258-272. · Zbl 1230.42020 · doi:10.1016/j.jmaa.2011.07.055
[14] J. Mateu, P. Mattila, A. Nicolau and J. Orobitg, BMO for nondoubling measures , Duke Math. J. 102 (2000), 533-565. · Zbl 0964.42009 · doi:10.1215/S0012-7094-00-10238-4
[15] F. Nazarov, S. Treil and A. Volberg, The $Tb$-theorem on non-homogeneous spaces , Acta Math. 190 (2003), 151-239. · Zbl 1065.42014 · doi:10.1007/BF02392690
[16] X. Tolsa, BMO, $H^1$ and Calderón-Zygmund operators for non doubling measures , Math. Ann. 319 (2001), 89-149. · Zbl 0974.42014 · doi:10.1007/s002080000144
[17] X. Tolsa, Littlewood-Paley theory and the $T(1)$ theorem with non-doubling measures , Adv. Math. 164 (2001), 57-116. · Zbl 1015.42010 · doi:10.1006/aima.2001.2011
[18] X. Tolsa, Painlevé’s problem and the semiadditivity of analytic capacity , Acta Math. 190 (2003), 105-149. · Zbl 1060.30031 · doi:10.1007/BF02393237