John-Nirenberg inequalities

Functions in Hardy spaces and in $\mathrm{BMO}$.[edit]

Let $D=\{z\in \mathbf{C}:|z|<1\}$ be the unit disc and let, for $1\le p<\mathrm{\infty }$, $H^p$ denote the space of holomorphic functions on $D$ (cf. also Analytic function) for which the supremum

$$\Vert f{\Vert }_{H^p}^p:=\frac{1}{2\pi }{\sup }_{r<1}{\int }_{-\pi }^{\pi }|f(r{e}^{i\vartheta }){|}^{p}d\vartheta$$

is finite. If a function $f$ belongs to $H^p$, $p\ge 1$, then there exists a function $f\in L^p(\partial D,d\vartheta /(2\pi ))$ such that

$$f(z)=\int k_{\vartheta }(z)f\left(e^{i\vartheta }\right)\frac{d\vartheta }{2\pi }.$$

Here, the function

$$e^{i\vartheta }\mapsto k_{\vartheta }(z)=\frac{1-|z{|}^2}{|z-e^{i\vartheta }{|}^2}$$

is the probability density (cf. also Density of a probability distribution) of a Brownian motion starting at $z\in D$ and exiting $D$ at $e^{i\vartheta }$. It is the Poisson kernel (cf. also Poisson integral) for the unit disc. A function $\phi$, defined on $[-\pi ,\pi ]$, belongs to $\mathrm{BMO}$ if there exists a constant $c$ such that ${\int }_I|\phi -{\phi }_I{|}^{2}d\vartheta \le c^2|I|$, for all intervals $I$ (cf. also $\mathrm{BMO}$-space). Here, ${\phi }_I={\int }_I\phi d\vartheta /|I|$ and $|I|$ denotes the Lebesgue measure of the interval $I$. Let ${\phi }_1$ and ${\phi }_2$ be bounded real-valued functions defined on the boundary $\partial D$ of $D$, and let ${\tilde{\phi }}_2$ be the boundary function of the harmonic conjugate function of the harmonic extension to $D$ of ${\phi }_2$ (cf. also Conjugate harmonic functions), so that ${\phi }_2+i{\tilde{\phi }}_2$ is the boundary function of a function which is holomorphic on $D$. Then the function ${\phi }_1+{\tilde{\phi }}_2$ belongs to $\mathrm{BMO}$: see [a4], p. 200, or [a9], p. 295. The function

$$\phi (\vartheta ):=\left|\log \left|\tan \frac{1}{2}\vartheta \right|\right|$$

belongs to $\mathrm{BMO}$, but is not bounded; see [a6], Chap. VI. Composition with the biholomorphic mapping

$$w\mapsto i\frac{1-w}{1+w}$$

turns $\mathrm{BMO}$-functions of the line into $\mathrm{BMO}$-functions of the circle; see [a6], p. 226.

Martingales in Hardy spaces and in $\mathcal{BMO}$.[edit]

Let $B_t$, $t\ge 0$, be Brownian motion starting at $0$ and let $\mathcal{F}$ be the filtration generated by Brownian motion (cf. also Stochastic processes, filtering of). Notice that $B_t$, $t\ge 0$, is a continuous Gaussian process with covariance $\mathsf{E}{B}_s{B}_t=\min (s,t)$. Define, for $0<p<\mathrm{\infty }$, the space of local martingales ${\mathcal{M}}^p$ by

$${\mathcal{M}}^p=\{X:\begin{array}{c}X\text{ a local martingale with respect to }\mathcal{F},\\ \mathsf{E}|X^{\ast }{|}^P<\mathrm{\infty }\end{array}\}.$$

Here, $X^{\ast }={\sup }_{t\ge 0}|X_t|$. Since the martingales are $\mathcal{F}$-martingales, they can be written in the form of an Itô integral:

$$X_t=X_0+{\int }_0^t{H}_s\cdot d{B}_s.$$

Here, $H$ is a predictable random process. Let $A$ be a $(2\times 2)$-matrix, and define the $A$-transform of $X$ by $(A^{\ast }X{)}_t={\int }_0^{t}A{H}_s.d{B}_s$. Then the martingale $X$ belongs to ${\mathcal{M}}^1$ if and only all transformed martingales $A\ast X$ have the property that

$${\sup }_{t>0}\mathsf{E}[|(A^{\ast }X{)}_t|]$$

is finite; this is Janson's theorem [a8]. A martingale $A\in {\mathcal{M}}^1$ is called an atom if there exists a stopping time $T$ such that

i) $A_t=0$ if $t\le T$; and

ii)

$$A^{\ast }={\sup }_{t\ge 0}|A_t|\le \frac{1}{\mathsf{P}[T<\mathrm{\infty }]}.$$

Since for atoms $A^{\ast }=0$ on the event $\{T=\mathrm{\infty }\}$, it follows that $\Vert A{\Vert }_1=\mathsf{E}[A^{\ast }]$. Moreover, every $X\in {\mathcal{M}}^1$ can be viewed as a limit of the form

$$X={\mathcal{M}}^1-{\lim }_{N\to \mathrm{\infty }}\sum _{n=-N}^{n=N}{c}_n{A}^n,$$

where every $A^n$ is an atom and where $\sum _{x\in \mathbf{N}}|c_n|\Vert X{\Vert }_1$. A local martingale $Y$ is said to have to bounded mean oscillation (notation $Y\in \mathcal{BMO}$) if there exists a constant $c$ such that

$$\mathsf{E}|Y_{\mathrm{\infty }}-Y_T|\le c\mathsf{P}[T<\mathrm{\infty }]$$

for all $\mathcal{F}$-stopping times $T$. The infimum of the constants $c$ is the $\mathcal{BMO}$-norm of $Y$. It is denoted by $\Vert Y{\Vert }_{\ast }$. The above inequality is equivalent to

$$\mathsf{E}[|Y_{\mathrm{\infty }}-Y_T||{\mathcal{F}}_T]\le c\text{almost surely}.$$

Let $X$ be a non-negative martingale. Put $X^{\ast }={\sup }_{s\ge 0}{X}_s$. Then $X$ belongs to ${\mathcal{M}}^1$ if and only if $\mathsf{E}[X_{\mathrm{\infty }}{\log }^{+}{X}_{\mathrm{\infty }}]$ is finite. More precisely, the following inequalities are valid:

$$\mathsf{E}[X_0]+\mathsf{E}\left[X_{\mathrm{\infty }}{\log }^{+}\frac{X_{\mathrm{\infty }}}{\mathsf{E}[X_0]}\right]\le$$

$$\le \mathsf{E}[X^{\ast }]\le$$

$$\le 2\mathsf{E}[X_0]+2\mathsf{E}\left[X_{\mathrm{\infty }}{\log }^{+}\frac{X_{\mathrm{\infty }}}{\mathsf{E}[X_0]}\right].$$

For details, see e.g. [a4], p. 149. Let $Y_t=B_{\min (t,1)}$. Then $Y$ is an unbounded martingale in $\mathcal{BMO}$. Two main versions of the John–Nirenberg inequalities are as follows.

Analytic version of the John–Nirenberg inequality.[edit]

There are constants $C$, $\gamma \in (0,\mathrm{\infty })$, such that, for any function $\phi \in \mathrm{BMO}$ for which $\Vert \phi {\Vert }_{\ast }\le 1$, the inequality

$$|\{\vartheta \in I:|\phi (e^{i\vartheta })-{\phi }_I|\ge \lambda \}|\le C{e}^{-\gamma \lambda }|I|$$

is valid for all intervals $I\subset [-\pi ,\pi ]$.

Probabilistic version of the John–Nirenberg inequality.[edit]

There exists a constant $C$ such that for any martingale $X\in {\mathcal{M}}^1$ for which $\Vert X\Vert \ast \le 1$, the inequality $\mathsf{P}[X^{\ast }>\lambda ]\le C{e}^{-\lambda /e}$ is valid. For the same constant $C$, the inequality

$$\mathsf{P}[{\sup }_{t\ge T}|X_t-X_T|>\lambda ]\le C{e}^{-\lambda /e}\mathsf{P}[T<\mathrm{\infty }]$$

is valid for all $\mathcal{F}$-stopping times $T$ and for all $X\in {\mathcal{M}}^1$ for which $\Vert X\Vert \ast \le 1$.

As a consequence, for $\phi \in \mathrm{BMO}$ integrals of the form ${\int }_{\partial D}\exp (\epsilon |\phi (e^{i\vartheta })-{\phi }_I|)d\vartheta$ are finite for $\epsilon >0$ sufficiently small.

Duality between $H^1$ and $\mathrm{BMO}$.[edit]

The John–Nirenberg inequalities can be employed to prove the duality between the space of holomorphic functions $H_0^1$ and $\mathrm{BMO}$ and between ${\mathcal{M}}^1$ and $\mathcal{BMO}$.

Duality between $H_0^1$ and $\mathrm{BMO}$ (analytic version).[edit]

The duality between $H_0^1=\{f\in H^1:f(0)=0\}$ and $\mathrm{BMO}$ is given by

$$(f,h)\mapsto {\int }_{\partial D}u(e^{i\vartheta })h(e^{i\vartheta })\frac{d\vartheta }{2\pi },$$

where $u(e^{i\vartheta })={\lim }_{r\uparrow 1}\mathrm{Re}f(r{e}^{i\vartheta })$ ($f\in H_0^1$, $h\in \mathrm{BMO}$).

Duality between ${\mathcal{M}}^1$ and $\mathcal{BMO}$ (probabilistic version).[edit]

Let $X$ be a martingale in ${\mathcal{M}}^1$ and let $Y$ be a martingale in $\mathcal{BMO}$. The duality between these martingales is given by $\mathsf{E}[X_{\mathrm{\infty }}{Y}_{\mathrm{\infty }}]$. Here, $X_{\mathrm{\infty }}={\lim }_{t\to \mathrm{\infty }}{X}_t$ and $Y_{\mathrm{\infty }}={\lim }_{t\to \mathrm{\infty }}{Y}_t$.

There exists a more or less canonical way to identify holomorphic functions in $H^1$ and certain continuous martingales in ${\mathcal{M}}^1$. Moreover, the same is true for functions of bounded mean oscillation (functions in $\mathrm{BMO}$) and certain continuous martingales in $\mathcal{BMO}$. Consequently, the duality between $H^1$ and $\mathrm{BMO}$ can also be extended to a duality between ${\mathcal{M}}^1$-martingales and $\mathcal{BMO}$-martingales.

The relationship between $H^1$ (respectively, $\mathrm{BMO}$) and a closed subspace of ${\mathcal{M}}^1$ (respectively, $\mathcal{BMO}$) is determined via the following equalities. For $f\in H^1$ one writes $u=\mathrm{Re}f$ and $U_t=u(B_{\min (t,\tau )})$, and for $h\in \mathrm{BMO}$ one writes $H_t=h(B_{\min (t,\tau )})$, where, as above, $B_t$ is two-dimensional Brownian motion starting at $0$, and where $\tau =\inf \{t>0:|B_t|=1\}$. Then the martingale $U$ belongs to ${\mathcal{M}}^1$, and $H$ is a member of $\mathcal{BMO}$. The fact that $H^1$ can be considered as a closed subspace of ${\mathcal{M}}^1$ is a consequence of the following

$$c\mathsf{E}\left[{\left|U_{\tau }^{\ast }\right|}^p\right]\le {\sup }_{0<r<1}{\int }_{\partial D}|f(r{e}^{i\vartheta }){|}^p\frac{d\vartheta }{2\pi }\le C\mathsf{E}\left[{\left|U_{\tau }^{\ast }\right|}^p\right],$$

$f\in H_0^p$, $U_t=\mathrm{Re}f(B_t)$, $U_{\tau }^{\ast }={\sup }_{0\le t<\tau }|U_t|$.

An important equality in the proof of these dualities is the following result: Let $f_1=u_1+i{v}_1$ and $f_2=u_2+i{v}_2$ be functions in $H_0^2$. Then

$$\mathsf{E}[U_{\mathrm{\infty }}^1{U}_{\mathrm{\infty }}^2]={\int }_{\partial D}{u}_1{u}_2\frac{d\vartheta }{2\pi }={\int }_{\partial D}{v}_1{v}_2\frac{d\vartheta }{2\pi }=\mathsf{E}[V_{\mathrm{\infty }}^1{V}_{\mathrm{\infty }}^2].$$

Here, $U_t^j=u_j(B_{\min (t,\tau )})$, $j=1,2$. A similar convention is used for $V_t^j$, $j=1,2$. In the first (and in the final) equality, the distribution of $\tau$ is used: $\mathsf{P}[\tau \in I]=|I|/(2\pi )$. The other equalities depend on the fact that a process like $U_t^1{U}_t^2-{\int }_0^t\mathrm{\nabla }{u}_1(B_s).\mathrm{\nabla }{u}_2(B_s)ds$ is a martingale, which follows from Itô calculus in conjunction with the harmonicity of the functions $u_1$ and $u_2$. Next, let $\phi$ be a function in $\mathrm{BMO}$. Denote by $h$ the harmonic extension of $\phi$ to $D$. Put $Y_t=h(B_{\min (t,\tau )})$. Then $Y_t$ is a continuous martingale. Let $T$ be any stopping time. From the Markov property it follows that $\mathsf{E}[|Y_{\mathrm{\infty }}-Y_T{|}^2|{\mathcal{F}}_T]=w(B_{\min (T,\tau )})$, where

$$w(z)=\int k_{\vartheta }(z)|\phi (e^{i\vartheta })-h(z){|}^2\frac{d\vartheta }{2\pi },$$

with

$$k_{\vartheta }(z)=\frac{1-|z{|}^2}{{|z-e^{i\vartheta }|}^2}.$$

As above, the Poisson kernel for the unit disc $e^{i\vartheta }\mapsto k_{\vartheta }(z)$ can be viewed as the probability density of a Brownian motion starting at $z\in D$ and exiting $D$ at $e^{i\vartheta }$. Since the inequality $w(z)\le c^2$ is equivalent to the inequality

$${\int }_I|\phi -{\phi }_I{|}^2\frac{d\vartheta }{2\pi }\le c_1^2|I|,$$

for some constant $c_1=c_1(c)$, it follows that $\mathrm{BMO}$ can be considered as a closed subspace of $\mathcal{BMO}$: 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 $u$ be function in $L^1(I)$ ($I$ is some interval). Suppose $|I|\alpha >{\int }_I|u(\vartheta )|d\vartheta$. Then there exists a pairwise disjoint sequence $\{I_j\}$ of open subintervals of $I$ such that $|u|\le \alpha$ almost everywhere on $I\mathrm{\setminus }\cup I_j$,

$$\alpha \le \frac{1}{|I_j|}{\int }_{I_j}|u(\vartheta )|d\vartheta <2\alpha ,$$

and

$$\sum |I_j|\le \frac{1}{\alpha }{\int }_I|u(\vartheta )|d\vartheta .$$

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 $\mathbf{C}$ (see [a6]), in ${\mathbf{R}}^d$ and in more general Riemannian manifolds ([a1], [a2], [a7], [a10]). For a relationship with Carleson measures, see [a6], Chap. 6. A measure $\lambda$ on $D$ is called a Carleson measure if $\lambda (S)\le K.h$ for some constant $K$ and for all circle sectors $S=\{r{e}^{i\vartheta }:1-h\le r<1,|\vartheta -{\vartheta }_0|\le h\}$. A function $\phi$ belongs to $\mathrm{BMO}$ if and only if

$$|\mathrm{\nabla }u(z){|}^2\log \frac{1}{|z|}dxdy$$

is a Carleson measure. Here, $u$ is the harmonic extension of $\phi$. For some other phenomena and related inequalities, see e.g. [a3], [a10], and [a11].

References[edit]