VMO-space

space of functions of vanishing mean oscillation

The class of functions of vanishing mean oscillation on $\mathbf R ^ {n}$, denoted by ${ \mathop{\rm VMO} } ( \mathbf R ^ {n} )$, is the subclass of ${ \mathop{\rm BMO} } ( \mathbf R ^ {n} )$ consisting of the functions $f$ with the property that

$${\lim\limits } _ {\left | Q \right | \rightarrow 0 } { \frac{1}{\left | Q \right | } } \int\limits _ { Q } {\left | {f - f _ {Q} } \right | } {dt } \rightarrow 0.$$

Here, $| Q |$ denotes the volume of the ball $Q$ and $f _ {Q}$ denotes the mean of $f$ over $Q$( see ${ \mathop{\rm BMO} }$- space). As with ${ \mathop{\rm BMO} }$, ${ \mathop{\rm VMO} }$ can be defined for spaces of homogeneous type.

Some properties of ${ \mathop{\rm VMO} }$ are as follows (see also [a1], [a2], [a3]). Bounded, uniformly continuous functions are in ${ \mathop{\rm VMO} }$( cf. Uniform continuity), and ${ \mathop{\rm VMO} }$ can be obtained as the ${ \mathop{\rm VMO} }$- closure of the continuous functions that vanish at infinity. The Hilbert transform of a bounded, uniformly continuous function on $\mathbf R$ is in ${ \mathop{\rm VMO} } ( \mathbf R )$. ${ \mathop{\rm VMO} } ( \mathbf R ^ {n} )$ is the dual of the Hardy space $H ^ {1} ( \mathbf R ^ {n} )$( cf. also Hardy spaces).

${ \mathop{\rm VMO} }$ appears in the theory of Douglas algebras: Let $T$ be the boundary of the unit disc in $\mathbf C$. Let $H ^ \infty$ denote the subspace of $L _ \infty ( T )$ consisting of the boundary values of bounded holomorphic functions (cf. Analytic function) on the unit disc and let $C$ denote the set of continuous functions on $T$. Put $H ^ \infty + C = \{ {f + g } : {f \in H ^ \infty , g \in C } \}$. $H ^ \infty + C$ is a closed subalgebra of $L _ \infty ( T )$ and the simplest example of a Douglas algebra. Its largest self-adjoint subalgebra, $QC$, equals $L _ \infty ( T ) \cap { \mathop{\rm VMO} } ( T )$, [a2], [a3].

See also ${ \mathop{\rm BMO} }$- 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=49101
This article was adapted from an original article by J. Wiegerinck (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article