John-Nirenberg inequalities
Functions in Hardy spaces and in
.
Let be the unit disc and let, for
,
denote the space of holomorphic functions on
(cf. also Analytic function) for which the supremum
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is finite. If a function belongs to
,
, then there exists a function
such that
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Here, the function
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is the probability density (cf. also Density of a probability distribution) of a Brownian motion starting at and exiting
at
. It is the Poisson kernel (cf. also Poisson integral) for the unit disc. A function
, defined on
, belongs to
if there exists a constant
such that
, for all intervals
(cf. also
-space). Here,
and
denotes the Lebesgue measure of the interval
. Let
and
be bounded real-valued functions defined on the boundary
of
, and let
be the boundary function of the harmonic conjugate function of the harmonic extension to
of
(cf. also Conjugate harmonic functions), so that
is the boundary function of a function which is holomorphic on
. Then the function
belongs to
: see [a4], p. 200, or [a9], p. 295. The function
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belongs to , but is not bounded; see [a6], Chap. VI. Composition with the biholomorphic mapping
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turns -functions of the line into
-functions of the circle; see [a6], p. 226.
Martingales in Hardy spaces and in
.
Let ,
, be Brownian motion starting at
and let
be the filtration generated by Brownian motion (cf. also Stochastic processes, filtering of). Notice that
,
, is a continuous Gaussian process with covariance
. Define, for
, the space of local martingales
by
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Here, . Since the martingales are
-martingales, they can be written in the form of an Itô integral:
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Here, is a predictable random process. Let
be a
-matrix, and define the
-transform of
by
. Then the martingale
belongs to
if and only all transformed martingales
have the property that
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is finite; this is Janson's theorem [a8]. A martingale is called an atom if there exists a stopping time
such that
i) if
; and
ii)
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Since for atoms on the event
, it follows that
. Moreover, every
can be viewed as a limit of the form
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where every is an atom and where
. A local martingale
is said to have to bounded mean oscillation (notation
) if there exists a constant
such that
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for all -stopping times
. The infimum of the constants
is the
-norm of
. It is denoted by
. The above inequality is equivalent to
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Let be a non-negative martingale. Put
. Then
belongs to
if and only if
is finite. More precisely, the following inequalities are valid:
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For details, see e.g. [a4], p. 149. Let . Then
is an unbounded martingale in
. Two main versions of the John–Nirenberg inequalities are as follows.
Analytic version of the John–Nirenberg inequality.
There are constants ,
, such that, for any function
for which
, the inequality
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is valid for all intervals .
Probabilistic version of the John–Nirenberg inequality.
There exists a constant such that for any martingale
for which
, the inequality
is valid. For the same constant
, the inequality
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is valid for all -stopping times
and for all
for which
.
As a consequence, for integrals of the form
are finite for
sufficiently small.
Duality between
and
.
The John–Nirenberg inequalities can be employed to prove the duality between the space of holomorphic functions and
and between
and
.
Duality between
and
(analytic version).
The duality between and
is given by
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where (
,
).
Duality between
and
(probabilistic version).
Let be a martingale in
and let
be a martingale in
. The duality between these martingales is given by
. Here,
and
.
There exists a more or less canonical way to identify holomorphic functions in and certain continuous martingales in
. Moreover, the same is true for functions of bounded mean oscillation (functions in
) and certain continuous martingales in
. Consequently, the duality between
and
can also be extended to a duality between
-martingales and
-martingales.
The relationship between (respectively,
) and a closed subspace of
(respectively,
) is determined via the following equalities. For
one writes
and
, and for
one writes
, where, as above,
is two-dimensional Brownian motion starting at
, and where
. Then the martingale
belongs to
, and
is a member of
. The fact that
can be considered as a closed subspace of
is a consequence of the following
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,
,
.
An important equality in the proof of these dualities is the following result: Let and
be functions in
. Then
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Here, ,
. A similar convention is used for
,
. In the first (and in the final) equality, the distribution of
is used:
. The other equalities depend on the fact that a process like
is a martingale, which follows from Itô calculus in conjunction with the harmonicity of the functions
and
. Next, let
be a function in
. Denote by
the harmonic extension of
to
. Put
. Then
is a continuous martingale. Let
be any stopping time. From the Markov property it follows that
, where
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with
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As above, the Poisson kernel for the unit disc can be viewed as the probability density of a Brownian motion starting at
and exiting
at
. Since the inequality
is equivalent to the inequality
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for some constant , it follows that
can be considered as a closed subspace of
: see [a6], Corol. 2.4; p. 234.
The analytic John–Nirenberg inequality can be viewed as a consequence of a result due to A.P. Calderón and A. Zygmund. Let be function in
(
is some interval). Suppose
. Then there exists a pairwise disjoint sequence
of open subintervals of
such that
almost everywhere on
,
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and
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In [a1], [a6], [a7] and [a10], extensions of the above can be found. In particular, some of the concepts can be extended to other domains in (see [a6]), in
and in more general Riemannian manifolds ([a1], [a2], [a7], [a10]). For a relationship with Carleson measures, see [a6], Chap. 6. A measure
on
is called a Carleson measure if
for some constant
and for all circle sectors
. A function
belongs to
if and only if
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is a Carleson measure. Here, is the harmonic extension of
. For some other phenomena and related inequalities, see e.g. [a3], [a10], and [a11].
References
[a1] | M. Biroli, U. Mosco, "Sobolev inequalities on homogeneous spaces: Potential theory and degenerate partial differential operators (Parma)" Potential Anal. , 4 (1995) pp. 311–324 |
[a2] | S.Y.A. Chang, R. Fefferman, "A continuous version of duality of ![]() ![]() |
[a3] | L. Chevalier, "Quelles sont les fonctions qui opèrent de ![]() ![]() ![]() ![]() |
[a4] | R. Durrett, "Brownian motion and martingales in analysis" , Wadsworth (1984) (Contains Mathematica analysis and stochastic processes) |
[a5] | J.B. Garnett, "Two constructions in ![]() |
[a6] | J. Garnett, "Bounded analytic functions" , Acad. Press (1981) |
[a7] | R. Hurri-Syrjanen, "The John–Nirenberg inequality and a Sobolev inequality in general domains" J. Math. Anal. Appl. , 175 : 2 (1993) pp. 579–587 |
[a8] | S. Janson, "Characterization of ![]() ![]() |
[a9] | P. Koosis, "Introduction to ![]() |
[a10] | Jia-Yu Li, "On the Harnack inequality for harmonic functions on complete Riemannian manifolds" Chinese Ann. Math. Ser. B , 14 : 1 (1993) pp. 1–12 |
[a11] | F.J. Martin–Reyes, A. de la Torre, "One-sided ![]() |
[a12] | G. Weiss, "Weak-type inequalities for ![]() ![]() |
John-Nirenberg inequalities. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=John-Nirenberg_inequalities&oldid=16042