Definition
Definition 1. The mean oscillation of a locally integrable function u (i.e. a function belonging to ) over a hypercube Q in ℝn is defined as the following integral:
where
- |Q| is the volume of Q, i.e. its Lebesgue measure
- uQ is the average value of u on the cube Q, i.e.
-
- .
Definition 2. A BMO function is any function u belonging to whose mean oscillation has a finite supremum over the set of all cubes Q contained in ℝn.
Note. The use of cubes Q in ℝn as the integration domains on which the mean oscillation is calculated, is not mandatory: Wiegerinck (2001) uses balls instead and, as remarked by Stein (1993, p. 140), in doing so a perfectly equivalent of definition of functions of bounded mean oscillation arises.
Read more about this topic: Bounded Mean Oscillation
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