- 1 Functions in Hardy spaces and in .
- 2 Martingales in Hardy spaces and in .
- 3 Duality between and .
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
is finite. If a function belongs to , , then there exists a function such that
Here, the function
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
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
Here, . Since the martingales are -martingales, they can be written in the form of an Itô integral:
i) if ; and
Since for atoms on the event , it follows that . Moreover, every can be viewed as a limit of the form
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
for all -stopping times . The infimum of the constants is the -norm of . It is denoted by . The above inequality is equivalent to
Let be a non-negative martingale. Put . Then belongs to if and only if is finite. More precisely, the following inequalities are valid:
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
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
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
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
, , .
An important equality in the proof of these dualities is the following result: Let and be functions in . Then
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
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
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 ,
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
|[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 with on the bidisc" Ann. of Math. (2) , 112 (1980) pp. 179–201|
|[a3]||L. Chevalier, "Quelles sont les fonctions qui opèrent de dans ou de dans " Bull. London Math. Soc. , 27 : 6 (1995) pp. 590–594|
|[a4]||R. Durrett, "Brownian motion and martingales in analysis" , Wadsworth (1984) (Contains Mathematica analysis and stochastic processes)|
|[a5]||J.B. Garnett, "Two constructions in " G. Weiss (ed.) S. Wainger (ed.) , Harmonic analysis in Euclidean spaces , Proc. Symp. Pure Math. , XXXV:1 , Amer. Math. Soc. (1979) pp. 295–301|
|[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 by singular integral transformations on martingales and " Math. Scand. , 41 (1977) pp. 140–152|
|[a9]||P. Koosis, "Introduction to -spaces: with an appendix on Wolff's proof of the corona theorem" , London Math. Soc. Lecture Notes , 40 , London Math. Soc. (1980)|
|[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 spaces" J. London Math. Soc. (2) , 49 : 3 (1994) pp. 529–542|
|[a12]||G. Weiss, "Weak-type inequalities for and " G. Weiss (ed.) S. Wainger (ed.) , Harmonic Analysis in Euclidean Spaces , Proc. Symp. Pure Math. , XXXV:1 , Amer. Math. Soc. (1979) pp. 295–301|
John-Nirenberg inequalities. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=John-Nirenberg_inequalities&oldid=16042