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Boundedness of Calderón-Zygmund operators on non-homogeneous metric measure spaces. (English) Zbl 1250.42044
Let $$(X,d, \mu)$$ be a separable metric space and let $$T$$ be a Calderón–Zygmund operator with standard kernel $$K$$: $Tf(x) := \int_X K(x,y)f(y)d\mu(y), \quad x \notin \text{supp}\;f.$ When $$\mu$$ satisfies the polynomial growth condition: $\mu( \{ y \in {\mathbb R}^n: | x-y | <r \}) \leq C r^a,$ 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 $$L^2(\mu)$$, then $$T$$ is bounded on $$L^p(\mu)$$ for all $$p \in (1,\infty)$$. The authors generalize this result as follows. If $$(X,d,\mu)$$ 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 $$\mu(\{ x \}) =0$$ for all $$\in X$$, then the boundedness of $$T$$ on $$L^2(\mu)$$ is equivalent to that of $$T$$ on $$L^p(\mu)$$ for some $$p \in (1,\infty)$$.

##### MSC:
 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory 30L99 Analysis on metric spaces
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