The spaces BMOH and BMOA When the
dimension of the ambient space is 1, the space BMO can be seen as a
linear subspace of
harmonic functions on the
unit disk and plays a major role in the theory of
Hardy spaces: by using , it is possible to define the BMO(
T) space on the
unit circle as the space of
functions f :
T →
R such that : \frac 1 \int_I|f(y)-f_I|\,\mathrm{d}y i.e. such that its over every arc I of the
unit circle is bounded. Here as before
fI is the mean value of f over the arc I. An Analytic function on the
unit disk is said to belong to the
Harmonic BMO or in the
BMOH space if and only if it is the
Poisson integral of a BMO(
T) function. Therefore, BMOH is the space of all functions
u with the form: : u(a) = \frac{1}{2\pi} \int_{\mathbf{T}}\frac{1-|a|^2}{\left|a-e^{i\theta}\right|^2} f(e^{i\theta})\,\mathrm{d}\theta equipped with the norm: :\|u\|_\text{BMOH}=\sup _ {|a| The subspace of analytic functions belonging BMOH is called the
Analytic BMO space or the
BMOA space.
BMOA as the dual space of H1(D) Charles Fefferman in his original work proved that the real BMO space is dual to the real valued harmonic Hardy space on the upper
half-space Rn × (0, ∞]. In the theory of Complex and Harmonic analysis on the unit disk, his result is stated as follows. Let
Hp(
D) be the Analytic
Hardy space on the
unit Disc. For
p = 1 we identify (
H1)* with BMOA by pairing
f ∈
H1(
D) and
g ∈ BMOA using the
anti-linear transformation Tg :T_g(f) = \lim_{r \to 1} \int_{-\pi}^\pi \bar{g}(e^{i\theta}) f(re^{i\theta}) \, \mathrm{d}\theta Notice that although the limit always exists for an
H1 function f and
Tg is an element of the dual space (
H1)*, since the transformation is
anti-linear, we don't have an isometric isomorphism between (
H1)* and BMOA. However one can obtain an isometry if they consider a kind of
space of conjugate BMOA functions.
The space VMO The space
VMO of functions of
vanishing mean oscillation is the closure in BMO of the continuous functions that vanish at infinity. It can also be defined as the space of functions whose "mean oscillations" on cubes
Q are not only bounded, but also tend to zero uniformly as the radius of the cube
Q tends to 0 or ∞. The space VMO is a sort of Hardy space analogue of the space of continuous functions vanishing at infinity, and in particular the real valued harmonic Hardy space
H1 is the dual of VMO.
Relation to the Hilbert transform A locally integrable function
f on
R is BMO if and only if it can be written as : f=f_1 + H f_2 + \alpha where
fi ∈
L∞, α is a constant and
H is the
Hilbert transform. The BMO norm is then equivalent to the infimum of \|f_1\|_\infty + \|f_2\|_\infty over all such representations. Similarly
f is VMO if and only if it can be represented in the above form with
fi bounded uniformly continuous functions on
R.
The dyadic BMO space Let
Δ denote the set of
dyadic cubes in
Rn. The space
dyadic BMO, written BMOd is the space of functions satisfying the same inequality as for BMO functions, only that the supremum is over all dyadic cubes. This supremum is sometimes denoted ||•||BMO
d. This space properly contains BMO. In particular, the function log(
x)
χ[0,∞) is a function that is in dyadic BMO but not in BMO. However, if a function
f is such that ||
f(•−
x)||BMOd ≤
C for all
x in
Rn for some
C > 0, then by the
one-third trick f is also in BMO. In the case of BMO on
Tn instead of
Rn, a function
f is such that ||
f(•−
x)||BMOd ≤
C for n+1 suitably chosen
x, then
f is also in BMO. This means BMO(
Tn ) is the intersection of n+1 translation of dyadic BMO. By duality, H1(
Tn ) is the sum of
n+1 translation of dyadic H1. Although dyadic BMO is a much narrower class than BMO, many theorems that are true for BMO are much simpler to prove for dyadic BMO, and in some cases one can recover the original BMO theorems by proving them first in the special dyadic case. ==Examples==