According to
Nirenberg (1985, p. 703 and p. 707),[1] the space of functions of bounded mean oscillation was introduced by
John (1961, pp. 410–411) in connection with his studies of
mappings from a
bounded setΩ belonging to Rn into Rn and the corresponding problems arising from
elasticity theory, precisely from the concept of
elastic strain: the basic notation was introduced in a closely following paper by
John & Nirenberg (1961),[2] where several properties of this function spaces were proved. The next important step in the development of the theory was the proof by
Charles Fefferman[3] of the
duality between BMO and the
Hardy spaceH1, in the noted paper
Fefferman & Stein 1972: a constructive proof of this result, introducing new methods and starting a further development of the theory, was given by
Akihito Uchiyama.[4]
Definition 2. A BMO function is a locally integrable function u whose mean oscillation
supremum, taken over the set of all
cubesQ contained in Rn, is finite.
Note 1. The supremum of the mean oscillation is called the BMO norm of u.[6] and is denoted by ||u||BMO (and in some instances it is also denoted ||u||∗).
The universally adopted notation used for the set of BMO functions on a given domain Ω is BMO(Ω): when Ω = Rn, BMO(Rn) is simply symbolized as BMO.
The BMO norm of a given BMO function u is denoted by ||u||BMO: in some instances, it is also denoted as ||u||∗.
Basic properties
BMO functions are locally p–integrable
BMO functions are locally Lp if 0 < p < ∞, but need not be locally bounded. In fact, using the John-Nirenberg Inequality, we can prove that
BMO is a Banach space
Constant functions have zero mean oscillation, therefore functions differing for a constant c > 0 can share the same BMO norm value even if their difference is not zero
almost everywhere. Therefore, the function ||u||BMO is properly a norm on the
quotient space of BMO functions
modulo the space of
constant functions on the domain considered.
Averages of adjacent cubes are comparable
As the name suggests, the mean or average of a function in BMO does not oscillate very much when computing it over cubes close to each other in position and scale. Precisely, if Q and R are
dyadic cubes such that their boundaries touch and the side length of Q is no less than one-half the side length of R (and vice versa), then
where C > 0 is some universal constant. This property is, in fact, equivalent to f being in BMO, that is, if f is a locally integrable function such that |fR−fQ| ≤ C for all dyadic cubes Q and R adjacent in the sense described above and f is in dyadic BMO (where the supremum is only taken over dyadic cubes Q), then f is in BMO.[7]
BMO is the dual vector space of H1
Fefferman (1971) showed that the BMO space is dual to H1, the
Hardy space with p = 1.[8] The pairing between f ∈ H1 and g ∈ BMO is given by
though some care is needed in defining this integral, as it does not in general converge absolutely.
The John–Nirenberg Inequality
The John–Nirenberg Inequality is an estimate that governs how far a function of bounded mean oscillation may deviate from its average by a certain amount.
Statement
For each , there are constants (independent of f), such that for any cube in ,
Conversely, if this inequality holds over all
cubes with some constant C in place of ||f||BMO, then f is in BMO with norm at most a constant times C.
A consequence: the distance in BMO to L∞
The John–Nirenberg inequality can actually give more information than just the BMO norm of a function. For a locally integrable function f, let A(f) be the infimal A>0 for which
The John–Nirenberg inequality implies that A(f) ≤ C||f||BMO for some universal constant C. For an
L∞ function, however, the above inequality will hold for all A > 0. In other words, A(f) = 0 if f is in L∞. Hence the constant A(f) gives us a way of measuring how far a function in BMO is from the subspace L∞. This statement can be made more precise:[9] there is a constant C, depending only on the
dimensionn, such that for any function f ∈ BMO(Rn) the following two-sided inequality holds
i.e. such that its mean oscillation over every arc I of the
unit circle[10] is bounded. Here as before fI is the mean value of f over the arc I.
Definition 3. 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:
equipped with the norm:
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-spaceRn × (0, ∞].[11] In the theory of Complex and Harmonic analysis on the unit disk, his result is stated as follows.[12] 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 transformationTg
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.[13]
Relation to the Hilbert transform
A locally integrable function f on R is BMO if and only if it can be written as
The BMO norm is then equivalent to the infimum of 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.[14]
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 ||•||BMOd.
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 trickf 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.[15]
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.[16]
Examples
Examples of BMO functions include the following:
All bounded (measurable) functions. If f is in L∞, then ||f||BMO ≤ 2||f||∞:[17] however, the converse is not true as the following example shows.
The function log(|P|) for any polynomial P that is not identically zero: in particular, this is true also for |P(x)| = |x|.[17]
If w is an
A∞ weight, then log(w) is BMO. Conversely, if f is BMO, then eδf is an A∞ weight for δ>0 small enough: this fact is a consequence of the
John–Nirenberg Inequality.[18]
Notes
^Aside with the collected papers of
Fritz John, a general reference for the theory of functions of bounded mean oscillation, with also many (short) historical notes, is the noted book by
Stein (1993, chapter IV).
Girela, Daniel (2001), "Analytic functions of bounded mean oscillation", in Aulaskari, Rauno (ed.), Complex function spaces, Proceedings of the summer school, Mekrijärvi, Finland, August 30-September 3, 1999, Univ. Joensuu Dept. Math. Rep. Ser., vol. 4,
Joensuu: Joensuu University, Department of Mathematics, pp. 61–170,
MR1820090,
Zbl0981.30026.