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Boundedness of Calderón-Zygmund operators on non-homogeneous metric measure spaces. (English) Zbl 1250.42044

Let (X,d,μ) be a separable metric space and let T be a Calderón–Zygmund operator with standard kernel K:

Tf(x):= X K(x,y)f(y)dμ(y),xsuppf·

When μ satisfies the polynomial growth condition:

μ({y n :|x-y|<r})Cr a ,

F. Nazarov, S. Treil and A. Volberg [Int. Math. Res. Not. 1998, No. 9, 463–487 (1998; Zbl 0918.42009)] proved that if T is bounded on L 2 (μ), then T is bounded on L p (μ) for all p(1,). The authors generalize this result as follows. If (X,d,μ) satisfies the upper doubling condition, the geometric doubling condition (see T. Hytönen [Publ. Mat., Barc. 54, No. 2, 485–504 (2010; Zbl 1246.30087)]) and the non-atomic condition that μ({x})=0 for all X, then the boundedness of T on L 2 (μ) is equivalent to that of T on L p (μ) for some p(1,).

42B20Singular and oscillatory integrals, several variables
42B25Maximal functions, Littlewood-Paley theory
30L99Complex analysis on metric spaces