Bounded Mean Oscillation - Definition

Definition

Definition 1. The mean oscillation of a locally integrable function u (i.e. a function belonging to ) over a hypercube Q in n 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 n.

Note. The use of cubes Q in n 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 definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.
    Samuel Taylor Coleridge (1772–1834)

    The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.
    William James (1842–1910)

    ... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.
    Adrienne Rich (b. 1929)