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
Famous quotes containing the word definition:
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)