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Local maximal operators on measure metric spaces. (English) Zbl 1291.42015

The paper deals with the notion of local maximal operators such as Hardy-Littlewood and the sharp maximal function and, consequently, the study of local BMO spaces. The locality condition consists of some restrictions on the radii of the balls involved. The hypothesis do not assume the doubling condition but some other geometrical restrictions are considered on the measure metric space.
The approach follows some ideas of the papers [A. Carbonaro et al., Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 8, No. 3, 543–582 (2009; Zbl 1180.42008); Colloq. Math. 118, No. 1, 13–41 (2010; Zbl 1193.42076)] where an approximate midpoint or isoperimetric properties were imposed. The study emphasizes the similarities and differences between local and global objects. A wide quantity of examples are provided within the paper that illustrate the different behaviour of these maximal operators for two different but equivalent metrics.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
51F99 Metric geometry
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