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