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:
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
“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 is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)