According to T. Hytönen [Publ. Mat., Barc. 54, No. 2, 485–504 (2010; Zbl 1246.30087)], a metric measure space is said to be upper doubling if is a Borel measure on and there exist a dominating function and a positive constant such that for each is non-decreasing and, for all and ,
Obviously, a space of homogeneous type is a special case of upper doubling spaces, where one can take the dominating function . Moreover, let be a non-negative Radon measure on which only satisfies the polynomial growth condition
By taking , we see that is also upper doubling measure space.
A metric space is said to be geometrically doubling if there exists some such that for any ball , there exists a finite ball covering of such that the cardinality of this covering is at most .
Let be a metric space satisfying the upper doubling condition and the geometrical doubling condition. T. Hytönen introduced the regularized BMO space RBMO. The authors introduce the atomic Hardy space and show that the dual space of is RBMO. As an application they obtain the boundedness of Calderón–Zygmund operators from to .