BMO-space
space of functions of bounded mean oscillation
Functions of bounded mean oscillation were introduced by F. John and L. Nirenberg [a8], [a12], in connection with differential equations. The definition on reads as follows: Suppose that
is integrable over compact sets in
, (i.e.
), and that
is any ball in
, with volume denoted by
. The mean of
over
will be
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By definition, belongs to
if
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where the supremum is taken over all balls . Here,
is called the
-norm of
, and it becomes a norm on
after dividing out the constant functions. Bounded functions are in
and a
-function is locally in
for every
. Typical examples of
-functions are of the form
with
a polynomial on
.
The space is very important in modern harmonic analysis. Taking
, the Hilbert transform
, defined by
, maps
to
boundedly, i.e.
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The same is true for a large class of singular integral transformations (cf. also Singular integral), including Riesz transformations [a12]. There is a version of the Riesz interpolation theorem (cf. also Riesz interpolation formula) for analytic families of operators ,
, which besides the
-boundedness assumptions on
involves the (weak) assumption
instead of the usual assumption
, cf. [a12]. However the most famous result is the Fefferman duality theorem, [a6], [a7], [a12]. It states that the dual of
is
. Here,
denotes the real Hardy space on
(cf. also Hardy spaces). The result is also valid for the usual space
on the disc or the upper half-plane, with an appropriate complex multiplication on
, cf. [a5].
Calderón–Zygmund operators on form an important class of singular integral operators. A Calderón–Zygmund operator can be defined as a linear operator
with associated Schwarz kernel
defined on
with the following properties:
i) is locally integrable on
and satisfies
;
ii) there exist constants and
such that for
and
,
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Similarly, for and
,
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iii) can be extended to a bounded linear operator on
.
This last condition is hard to verify in general. Thus, it is an important result, known as the -theorem, that if i) and ii) hold, then iii) is equivalent to:
is weakly bounded on
and both
and
are in
, cf. [a3], [a11], [a12]. It is known that diagonal operators with respect to an orthonormal wavelet basis are of Calderón–Zygmund type. This connection with wavelet analysis is treated in [a11].
Many of the results concerning -functions have been generalized to the setting of martingales, cf. [a9] (see also Martingale).
The duality result indicates that plays a role in complex analysis as well. The class of holomorphic functions (cf. Analytic function) on a domain
with boundary values in
is denoted by
, and is called the
-space, i.e.,
.
Carleson's corona theorem [a5] for the disc states that for given bounded holomorphic functions such that
there exist bounded holomorphic functions
such that
. So far (1996), this result could not be extended to the unit ball in
,
, but it can be proved if one only requires that
, cf. [a13].
The definition of makes sense as soon as there are proper notions of integral and ball in a space. Thus,
can be defined in spaces of homogeneous type, cf. [a1], [a2], [a10]. In the setting of several complex variables, several types of
-spaces arise on the boundary of (strictly) pseudoconvex domains, depending on whether one considers the isotropic Euclidean balls or the non-isotropic balls that are natural in connection with pseudo-convexity, cf. [a10].
References
[a1] | R.R. Coifman, G. Weiss, "Analyse harmonique non-commutative sur certains espaces homogènes" , Lecture Notes in Mathematics , 242 , Springer (1971) |
[a2] | R.R. Coifman, G. Weiss, "Extensions of Hardy spaces and their use in analysis" Bull. Amer. Math. Soc. , 83 (1977) pp. 569–643 |
[a3] | G. David, J.-L. Journé, "A boundedness criterion for generalized Calderón–Zygmund operators" Ann. of Math. , 120 (1985) pp. 371–397 |
[a4] | J. Garcia-Cuervas, J.L. Rubio de Francia, "Weighted norm inequalities and related topics" , Math. Stud. , 116 , North-Holland (1985) |
[a5] | J. Garnett, "Bounded analytic functions" , Acad. Press (1981) |
[a6] | C. Fefferman, "Characterizations of bounded mean oscillation" Bull. Amer. Math. Soc. , 77 (1971) pp. 587–588 |
[a7] | C. Fefferman, E.M. Stein, "![]() |
[a8] | F. John, L. Nirenberg, "On functions of bounded mean oscillation" Comm. Pure Appl. Math. , 14 (1961) pp. 415–426 |
[a9] | N. Kazamaki, "Continuous exponential martingales and BMO" , Lecture Notes in Mathematics , 579 , Springer (1994) |
[a10] | S.G. Krantz, "Geometric analysis and function spaces" , CBMS , 81 , Amer. Math. Soc. (1993) |
[a11] | Y. Meyer, "Ondelettes et opérateurs II. Opérateurs de Calderón–Zygmund" , Actual. Math. , Hermann (1990) |
[a12] | E.M. Stein, "Harmonic analysis: real variable methods, orthogonality, and oscillatory integrals" , Math. Ser. , 43 , Princeton Univ. Press (1993) |
[a13] | N.Th. Varopoulos, "BMO functions and the ![]() |
BMO-space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=BMO-space&oldid=19103