Let be a separable metric space and let be a Calderón–Zygmund operator with standard kernel :
When satisfies the polynomial growth condition:
F. Nazarov, S. Treil and A. Volberg [Int. Math. Res. Not. 1998, No. 9, 463–487 (1998; Zbl 0918.42009)] proved that if is bounded on , then is bounded on for all . The authors generalize this result as follows. If 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 for all , then the boundedness of on is equivalent to that of on for some .