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Non-homogeneous \(Tb\) theorem and random dyadic cubes on metric measure spaces. (English) Zbl 1261.42017
F. Nazarov, S. Treil and A. Volberg [Acta Math. 190, No. 2, 151–239 (2003; Zbl 1065.42014)] proved a \(Tb\) theorem on a metric measure spaces \((X,d,\mu)\) which satisfies the upper power bound condition \(\mu (B(x,r)) \leq Cr^d\).
T. Hytönen [Publ. Mat., Barc. 54, No. 2, 485–504 (2010; Zbl 1246.30087)] introduced quasimetric measure spaces equipped with an upper doubling measure. This is a property that encompasses both the doubling measure and the upper power bound condition. He developed a theory of BMO on these spaces.
The authors prove a \(Tb\) theorem on quasimetric spaces equipped with upper doubling measures. For the proof, they use a variant of system of dyadic cubes due to M. Christ [Colloq. Math. 60/61, No. 2, 601–628 (1990; Zbl 0758.42009)].

MSC:
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
30L99 Analysis on metric spaces
60D05 Geometric probability and stochastic geometry
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