Namespaces
Variants
Actions

VMO-space

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.


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