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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 \(p \in (1,\infty)\).

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