# VMO-space

*space of functions of vanishing mean oscillation*

The class of functions of vanishing mean oscillation on , denoted by , is the subclass of consisting of the functions with the property that

Here, denotes the volume of the ball and denotes the mean of over (see -space). As with , can be defined for spaces of homogeneous type.

Some properties of are as follows (see also [a1], [a2], [a3]). Bounded, uniformly continuous functions are in (cf. Uniform continuity), and can be obtained as the -closure of the continuous functions that vanish at infinity. The Hilbert transform of a bounded, uniformly continuous function on is in . is the dual of the Hardy space (cf. also Hardy spaces).

appears in the theory of Douglas algebras: Let be the boundary of the unit disc in . Let denote the subspace of consisting of the boundary values of bounded holomorphic functions (cf. Analytic function) on the unit disc and let denote the set of continuous functions on . Put . is a closed subalgebra of and the simplest example of a Douglas algebra. Its largest self-adjoint subalgebra, , equals , [a2], [a3].

See also -space.

#### References

[a1] | R.R. Coifman, G. Weiss, "Extensions of Hardy-spaces and their use in analysis" Bull. Amer. Math. Soc. , 83 (1977) pp. 569–645 |

[a2] | J. Garnett, "Bounded analytic functions" , Acad. Press (1981) |

[a3] | D. Sarason, "Functions of vanishing mean oscillation" Trans. Amer. Math. Soc. , 207 (1975) pp. 391–405 |

**How to Cite This Entry:**

VMO-space.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=VMO-space&oldid=18125