A remark on John – Nirenberg theorem for martingales
This paper is mainly devoted to establishing an extension of the John – Nirenberg theorem for martingales, more precisely, let $1 < p < \infty$ and $0 < q < \infty$. If the stochastic basis $(\scr {F_n})_n\geq 0$ is regular, then $BMO_{p,q} = BMO_1$ with the equivalent n...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507489008615424 |
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| author | Li, L. Лі, Л. |
| author_facet | Li, L. Лі, Л. |
| author_sort | Li, L. |
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| description | This paper is mainly devoted to establishing an extension of the John – Nirenberg theorem for martingales, more precisely,
let $1 < p < \infty$ and $0 < q < \infty$. If the stochastic basis $(\scr {F_n})_n\geq 0$ is regular, then $BMO_{p,q} = BMO_1$ with the equivalent
norms. Our method is to use a new atomic decomposition construction of the martingale Hardy space. |
| first_indexed | 2026-03-24T02:10:07Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 519.21
L. Li (College of Mathematics and Econometrics, Hunan Univ., China)
A REMARK ON JOHN – NIRENBERG THEOREM FOR MARTINGALES
ЗАУВАЖЕННЯ ЩОДО ТЕОРЕМИ ДЖОНА – НIРЕНБЕРГА
ДЛЯ МАРТИНГАЛIВ
This paper is mainly devoted to establishing an extension of the John – Nirenberg theorem for martingales, more precisely,
let 1 < p < \infty and 0 < q < \infty . If the stochastic basis (\scrF n)n\geq 0 is regular, then BMOp,q = BMO1 with the equivalent
norms. Our method is to use a new atomic decomposition construction of the martingale Hardy space.
Роботу, в основному, присвячено доведенню узагальнення теореми Джона – Нiренберга для мартингалiв, бiльш
точно, для 1 < p < \infty та 0 < q < \infty . За умови, що стохастичний базис (\scrF n)n\geq 0 є регулярним, маємо BMOp,q =
= BMO1 з еквiвалентними нормами. Наш метод зводиться до застосування нової конструкцiї атомного розкладу
простору мартингалiв Гардi.
1. Introduction. The John – Nirenberg theorem has been successfully extended to different settings
in recent years. A lot of works have been done on this subject (see [5, 6, 8 – 11, 19, 20]).
This remark deals with the John – Nirenberg theorem on Lorentz space for the martingale setting.
Before describing our main results, we recall the classical John – Nirenberg theorem in the martingale
theory. Let (\Omega ,\scrF ,\BbbP ) be a probability space, and \{ \scrF n\} n\geq 0 be a nondecreasing sequence of sub-\sigma -
algebras of \scrF such that \scrF = \sigma (
\bigcup
n\geq 0\scrF n). The expectation operator and the conditioned expectation
operator are denoted by \BbbE and \BbbE n, respectively. A sequence f = (fn)n\geq 0 of random variables such
that fn is \scrF n-measurable is said to be a martingale if \BbbE (| fn| ) < \infty and \BbbE n(fn+1) = fn for every
n \geq 0. We always suppose that for a martingale f, f0 = 0. The Banach spaces BMOp, 1 \leq p < \infty
are defined as follows:
BMOp =
\biggl\{
f = (fn)n\geq 0 : \| f\| BMOp = \mathrm{s}\mathrm{u}\mathrm{p}
n
\| \BbbE n (| f - fn| p) \|
1
p
\infty < \infty
\biggr\}
.
Here the f in | f - fn| p means f\infty . It can be shown that \| f\| BMOp admits an alternative definition
\| f\| BMOp = \mathrm{s}\mathrm{u}\mathrm{p}
\tau \in \scrT
\bigm\| \bigm\| (f - f \tau )\chi \{ \tau <\infty \}
\bigm\| \bigm\|
p
\| \chi \{ \tau <\infty \} \| p
,
where \scrT denotes the set of all stopping times with respect to \{ \scrF n\} n\geq 0. The well-known John –
Nirenberg theorem (see [13, 18]) says that if the stochastic basis \{ \scrF n\} n\geq 0 is regular, then
BMOp = BMO1.
In 2014, Yi, Wu and Jiao [19] extended this result to a wider class of the rearrangement invariant
Banach function space. That is, let E be a rearrangement invariant Banach function space on \Omega with
upper Boyd indices qE < \infty and define
BMOE = \{ f = (fn)n\geq 0 : \| f\| BMOE
< \infty \} ,
where
c\bigcirc L. Li, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11 1571
1572 L. LI
\| f\| BMOE
= \mathrm{s}\mathrm{u}\mathrm{p}
\nu \in \scrT
\bigm\| \bigm\| (f - f\nu )\chi \{ \nu <\infty \}
\bigm\| \bigm\|
E\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
E
.
Then if the stochastic basis is regular,
BMOE = BMO1.
Hence it is natural to consider whether the John – Nirenberg theorem is true for the nonrearrange-
ment invariant Banach function space. We will work on this problem in the present paper. Our goal
is to establish the John – Nirenberg theorem in the context of Lorentz spaces Lp,q, 1 < p < \infty ,
0 < q \leq 1. Note that such spaces are not the rearrangement invariant Banach function spaces. The
following is one of our main results:
Theorem 1.1. Let 1 < p < \infty and 0 < q < \infty . If the stochastic basis (\scrF n)n\geq 0 is regular, then
BMOp,q = BMO1 with equivalent norms,
where
BMOp,q =
\bigl\{
f = (fn)n\geq 0 : \| f\| BMOp,q < \infty
\bigr\}
,
and
\| f\| BMOp,q = \mathrm{s}\mathrm{u}\mathrm{p}
\nu \in \scrT
\bigm\| \bigm\| (f - f\nu )\chi \{ \nu <\infty \}
\bigm\| \bigm\|
p,q\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
p,q
.
Our main method is to use a new atomic decomposition construction of Hardy spaces by atoms
associated with Lorentz spaces.
2. Preliminaries. In this section, we give some preliminaries necessary for the whole paper. Let
us first recall some basic facts on the Lorentz spaces. Let (\Omega ,\scrF ,\BbbP ) be a complete probability space
and f be a measurable function defined on \Omega . The distribution function of f is the function \lambda s(f)
defined by
\lambda s(f) = \BbbP
\bigl( \bigl\{
\omega \in \Omega : | f(\omega )| > s
\bigr\} \bigr)
, s \geq 0.
And denote by \mu t(f) the decreasing rearrangement of f, defined by
\mu t(f) = \mathrm{i}\mathrm{n}\mathrm{f} \{ s \geq 0 : \lambda s(f) \leq t\} , t \geq 0,
with the convention that \mathrm{i}\mathrm{n}\mathrm{f} \varnothing = \infty .
The Lorentz space Lp,q(\Omega ,\scrF ,\BbbP ), 0 < p < \infty , 0 < q \leq \infty , consists of the measurable functions
f with finite norm or quasinorm \| f\| p,q given by
\| f\| p,q =
\left( q
p
\infty \int
0
\Bigl(
t
1
p\mu t(f)
\Bigr) q dt
t
\right) 1
q
, 0 < q < \infty ,
\| f\| p,\infty = \mathrm{s}\mathrm{u}\mathrm{p}
t>0
t
1
p\mu t(f), q = \infty .
It will be convenient for us to use the equivalent definition of \| f\| p,q, known as
\| f\| p,q =
\left( q
\infty \int
0
\Bigl(
t\BbbP (| f(x)| > t)
1
p
\Bigr) q dt
t
\right) 1
q
, 0 < q < \infty ,
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
A REMARK ON JOHN – NIRENBERG THEOREM FOR MARTINGALES 1573
\| f\| p,\infty = \mathrm{s}\mathrm{u}\mathrm{p}
t>0
t\BbbP (| f(x)| > t)
1
p , q = \infty .
These spaces are the generalizations of ordinary Lp spaces and they coincide with Lp when q = p.
As we known, if 1 < p < \infty and 1 \leq q \leq \infty , or p = q = 1, then \| \cdot \| p,q is equivalent to a norm.
However, for the other values of p and q, \| \cdot \| p,q is only a quasinorm. In particular, if 0 < q \leq 1
and q \leq p < \infty , then \| \cdot \| p,q is equivalent to a q-norm. The following lemmas can be found in
Grafakos [1].
Lemma 2.1. Let 0 < p, p1, p2 < \infty and 0 < q, p\prime 1, p
\prime
2 \leq \infty with 1/p = 1/p1 + 1/p\prime 1 and
1/q = 1/p2 + 1/p\prime 2, then
\| fg\| p,q \leq C\| f\| p1,p2\| g\| p\prime 1,p\prime 2 .
Moreover, if p = q, p1 = q1 and p2 = q2, we have
\| fg\| p \leq \| f\| p1\| g\| p\prime 1 .
Lemma 2.2. Let 1 < p < \infty and 0 < q \leq 1 with 1 = 1/p+ 1/p\prime , then the dual space of Lp,q
is Lp\prime ,\infty .
Now we define the Hardy martingale spaces. For a martingale f = (fn)n\geq 0, the maximal
function of martingale f is defined by
Mn(f) = \mathrm{s}\mathrm{u}\mathrm{p}
1\leq i\leq n
| fi| , M(f) = \mathrm{s}\mathrm{u}\mathrm{p}
i\geq 0
| fi| .
Define
H\ast
p =
\Bigl\{
f = (fn)n\geq 0 : \| f\| H\ast
p
=
\bigm\| \bigm\| M(f)
\bigm\| \bigm\|
p
< \infty
\Bigr\}
, 0 < p < \infty ,
H\ast
p,q =
\Bigl\{
f = (fn)n\geq 0 : \| f\| H\ast
p,q
=
\bigm\| \bigm\| M(f)
\bigm\| \bigm\|
p,q
< \infty
\Bigr\}
, 0 < p < \infty , 0 < q \leq \infty .
The stochastic basis (\scrF n)n\geq 0 is said to be regular, if for n \geq 1 and A \in \scrF n, there exists a
B \in \scrF n - 1 such that A \subset B and \BbbP (B) \leq R\BbbP (A), where R is a positive constant independent of n.
A martingale is said to be regular if it is adapted to a regular \sigma -algebra sequence. This amounts to
saying that there exists a constant R > 0 such that
fn \leq Rfn - 1
for all non-negative martingales (fn)n\geq 0 adapted to the stochastic basis (\scrF n)n\geq 0. We refer to Long
[13] and Weisz [18] for the theory of martingale Hardy spaces.
3. Main results. In this section we present the new John – Nirenberg theorem by constructing
the atomic decomposition of Hardy spaces H\ast
p via atoms associated with Lq,\infty -space for 1 < q < \infty .
We refer to [2 – 4, 7, 14, 17] for more information on the classical atomic decompositions.
Definition 3.1. Let 0 < p < \infty and 1 < q < \infty . A measurable function, a, is called a
(p, Lq,\infty )-atom if there exists a stopping time \nu such that
(1) an = Ena = 0 if \nu \geq n,
(2)
\bigm\| \bigm\| M(a)
\bigm\| \bigm\|
q,\infty \leq
\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
q,\infty
\BbbP (\nu < \infty )1/p
.
We denote the set of (p, Lq,\infty ) atoms by \scrA p,Lq,\infty .
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
1574 L. LI
Theorem 3.1. Let the stochastic basis (\scrF n)n\geq 0 be regular and 0 < p \leq 1 < q < \infty . Then
f \in H\ast
p if and only if there exist a sequence (ak) of (p, Lq,\infty ) atoms and a sequence (\mu k) \in \ell p of
real numbers such that
f =
\sum
k\in Z
\mu ka
k a.e.,
and
\| f\| H\ast
p
\approx \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
\| (\mu k)\| \ell p
\bigr\}
,
where the infimum is taken over all the preceding decompositions of f.
The proof of Theorem 3.1 uses the following well known lemma which is proved in Theo-
rem 7.1.2 of [13, p. 265].
Lemma 3.1. If the stochastic basis (\scrF n)n\geq 0 is regular, then for all non-negative adapted pro-
cesses \gamma = (\gamma n)n\geq 0 and \lambda \geq \| \gamma 0\| \infty , there exist a constant R > 0 and a stopping time \tau \lambda such
that
\{ M(\gamma ) > \lambda \} \subseteq \{ \tau \lambda < \infty \} ,
\BbbP (\tau \lambda < \infty ) \leq R\BbbP (M(\gamma ) > \lambda ),
\mathrm{s}\mathrm{u}\mathrm{p}
n\leq \tau \lambda
\gamma n = M\tau \lambda (\gamma ) \leq \lambda ,
\| \gamma 0\| \infty \leq \lambda 1 \leq \lambda 2 implies \tau \lambda 1 \leq \tau \lambda 2 .
Proof of Theorem 3.1. Let f \in H\ast
p . For the process (| fn| )n\geq 0 and \lambda k = 2k, define the stopping
time \tau k associate with \lambda k satisfying the Lemma 3.1. Since \{ \tau k\} is increasing and \BbbP (\tau k < \infty ) \rightarrow 0
as k \rightarrow \infty , we see \mathrm{l}\mathrm{i}\mathrm{m}k\rightarrow \infty \tau k = \infty , a.e.,
\mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
f\tau k = f a.e. and \mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow - \infty
| f\tau k | \leq \mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow - \infty
2k = 0 a.e.
Therefore, we get the following decomposition which converges pointwise:
fn =
\infty \sum
k= - \infty
\bigl(
f \tau k
n - f
\tau k - 1
n
\bigr)
\forall n \geq 0.
Set \mu k = 2k+1\BbbP (\tau k - 1 < \infty )1/p for all k \in \BbbZ . When \mu k \not = 0, we define
akn =
f \tau k
n - f
\tau k - 1
n
\mu k
\forall n \geq 0.
If \mu k = 0, then let akn = 0 for all k \in \BbbZ , n \in \BbbN . Then (akn)n\geq 0 is a martingale for each fixed k \in \BbbZ .
Since M\tau k(f) \leq 2k, we obtain
M
\Bigl(
akn
\Bigr)
\leq M (f \tau k) +M (f \tau k - 1)
\mu k
\leq \BbbP (\tau k - 1 < \infty ) - 1/p.
Hence it is easy to check that
\bigl(
akn
\bigr)
n\geq 0
is a bounded L2-martingale. Consequently, there exists an
element ak \in L2 such that \BbbE na
k = a.nk If n \leq \tau k - 1, then akn = 0, and
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
A REMARK ON JOHN – NIRENBERG THEOREM FOR MARTINGALES 1575
\bigm\| \bigm\| M(ak)
\bigm\| \bigm\|
q,\infty \leq
\bigm\| \bigm\| M(ak)
\bigm\| \bigm\|
\infty \| \chi \{ \tau k - 1<\infty \} \| q,\infty \leq
\| \chi \{ \tau k - 1<\infty \} \| q,\infty
\BbbP (\tau k - 1 < \infty )1/p
.
Thus we conclude that
\bigl(
ak, \tau k - 1
\bigr)
is really a (p, Lq,\infty )-atom. In view of Lemma 3.1, we have\sum
k\in \BbbZ
\mu p
k =
\sum
k\in \BbbZ
2(k+1)p\BbbP (\tau k - 1 < \infty ) \leq R
\sum
k\in \BbbZ
2(k+1)p\BbbP
\Bigl(
M(f) > 2k - 1
\Bigr)
\leq
\leq 8ppR
\sum
k\in \BbbZ
2(k - 1)p\int
2(k - 2)p
tp\BbbP (M(f) > t) dt = 8ppR\| f\| pH\ast
p
.
For the converse part, it suffices to prove that for any a \in \scrA p,Lq,\infty ,
\| a\| H\ast
p
=
\bigm\| \bigm\| M(a)
\bigm\| \bigm\|
p
\leq C.
Indeed, for 0 < p \leq 1,
\| f\| H\ast
p
=
\bigm\| \bigm\| M(f)
\bigm\| \bigm\|
p
\leq
\Biggl( \sum
k\in Z
\bigm\| \bigm\| \bigm\| \mu kM(ak)
\bigm\| \bigm\| \bigm\| p
p
\Biggr) 1/p
=
\Biggl( \sum
k\in Z
| \mu k| p
\bigm\| \bigm\| \bigm\| M(ak)
\bigm\| \bigm\| \bigm\| p
p
\Biggr) 1/p
.
We first consider the case 0 < p < 1. Given a (p, Lq,\infty )-atom a, we get\bigm\| \bigm\| M(a)
\bigm\| \bigm\|
1
=
\bigm\| \bigm\| M(a\chi \{ \nu <\infty \} )
\bigm\| \bigm\|
1
\leq C
\bigm\| \bigm\| M(a)
\bigm\| \bigm\|
q,\infty
\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
q\prime ,1
\leq
\leq C
\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
q,\infty \BbbP (\nu < \infty ) - 1/p
\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
q\prime ,1
= C\BbbP (\nu < \infty )1 - 1/p,
where \nu is the stopping time corresponding to a. Note that 1 +
1
p/(1 - p)
=
1
p
, we obtain
\bigm\| \bigm\| M(a)
\bigm\| \bigm\|
p
=
\bigm\| \bigm\| M(a\chi \{ \nu <\infty \} )
\bigm\| \bigm\|
p
\leq
\bigm\| \bigm\| M(a)
\bigm\| \bigm\|
1
\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
p
1 - p
\leq
\leq C\BbbP (\nu < \infty )1 - 1/p\BbbP (\nu < \infty )
1 - p
p = C.
As for the case p = 1, we directly have\bigm\| \bigm\| M(a)
\bigm\| \bigm\|
1
=
\bigm\| \bigm\| M(a\chi \{ \nu <\infty \} )
\bigm\| \bigm\|
1
\leq C
\bigm\| \bigm\| M(a)
\bigm\| \bigm\|
q,\infty
\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
q\prime ,1
\leq
\leq C
\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
q,\infty \BbbP (\nu < \infty ) - 1
\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
q\prime ,1
= C.
Theorem 3.1 is proved.
Theorem 3.2. Let 1 < p < \infty and 0 < q \leq 1. If the stochastic basis (\scrF n)n\geq 0 is regular, then
BMOp,q = BMO1 in the sense of equivalent norm.
Before proving Theorem 3.2, we present the maximal inequality for the martingale Lorentz –
Hardy spaces.
Lemma 3.2 (see [12]). Let f = (fn)n\geq 0 \in Lq,\infty , 1 < q < \infty , then there exists a constant Cq
(depending only on q) such that
\| f\| q,\infty \leq \| f\| H\ast
q,\infty \leq Cq\| f\| q,\infty .
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
1576 L. LI
Proof of Theorem 3.2. First suppose that f \in BMOp,q, then by Lemma 2.2 we have
\| f\| BMO1 = \mathrm{s}\mathrm{u}\mathrm{p}
\nu \in \scrT
\bigm\| \bigm\| (f - f\nu )\chi \{ \nu <\infty \}
\bigm\| \bigm\|
1
\BbbP (\nu < \infty )
\leq \mathrm{s}\mathrm{u}\mathrm{p}
\nu \in \scrT
C
\bigm\| \bigm\| (f - f\nu )\chi \{ \nu <\infty \}
\bigm\| \bigm\|
p,q
\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
p\prime ,\infty
\BbbP (\nu < \infty )
=
= C \mathrm{s}\mathrm{u}\mathrm{p}
\nu \in \scrT
\bigm\| \bigm\| (f - f\nu )\chi \{ \nu <\infty \}
\bigm\| \bigm\|
p,q\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
p,q
= C\| f\| BMOp,q .
On the other hand, assume that f \in BMO1, then from Lemma 2.1 and the definition of supremum,
there exists a function g \in Lp\prime ,\infty with \| g\| Lp\prime ,\infty \leq 1 such that
\bigm\| \bigm\| (f - f\nu )\chi \{ \nu <\infty \}
\bigm\| \bigm\|
p,q
\leq C1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
\{ \nu <\infty \}
(f - f\nu )g d\BbbP
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| .
According to Lemma 3.2, there exists a constant Cp\prime such that\bigm\| \bigm\| M(f)
\bigm\| \bigm\|
p\prime ,\infty \leq Cp\prime \| f\| p\prime ,\infty \forall f \in Lp\prime ,\infty .
Let
a =
\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
p\prime ,\infty (g - g\nu )
2Cp\prime \BbbP (\nu < \infty )
.
Then we obtain\bigm\| \bigm\| M(a)
\bigm\| \bigm\|
p\prime ,\infty \leq Cp\prime \| a\| p\prime ,\infty =
\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
p\prime ,\infty
2\BbbP (\nu < \infty )
\| g - g\nu \| p\prime ,\infty \leq
\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
p\prime ,\infty
\BbbP (\nu < \infty )
,
which means a \in \scrA 1,Lp\prime ,\infty . Then it follows from Theorem 3.1 that a \in H\ast
1 and \| a\| H\ast
1
= 1. Thus
g - g\nu =
2Cp\prime \BbbP (\nu < \infty )\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
p\prime ,\infty
a \in H\ast
1 ,
with its norm
\| g - g\nu \| H\ast
1
\leq
2Cp\prime \BbbP (\nu < \infty )\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
p\prime ,\infty
.
Since the stochastic basis (\scrF n)n\geq 0 is regular, the dual space of H\ast
1 is BMO1 (see [17, 20]). Hence
\bigm\| \bigm\| (f - f\nu )\chi \{ \nu <\infty \}
\bigm\| \bigm\|
p,q\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
p,q
\leq
C1
\bigm| \bigm| \bigm| \bigm| \bigm|
\int
\{ \nu <\infty \}
(f - f\nu )gd\BbbP
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
p,q
=
=
C1
\bigm| \bigm| \bigm| \bigm| \bigm|
\int
\{ \nu <\infty \}
f (g - g\nu ) d\BbbP
\bigm| \bigm| \bigm| \bigm| \bigm|
\| \chi \{ \nu <\infty \} \| p,q
\leq C1C2 \| g - g\nu \| H\ast
1
\| f\| BMO1
1\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
p,q
\leq
\leq 2C1C2Cp\prime
\BbbP (\nu < \infty )\| f\| BMO1\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
p,q
\bigm\| \bigm\| \chi \{ \nu <\infty \}
\bigm\| \bigm\|
p\prime ,\infty
= C\| f\| BMO1 .
Here C = 2C1C2Cp\prime . This means \| f\| BMOp,q \leq C\| f\| BMO1 .
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
A REMARK ON JOHN – NIRENBERG THEOREM FOR MARTINGALES 1577
Corollary 1. Let 1 < p < \infty and 0 < q < \infty . If the stochastic basis (\scrF n)n\geq 0 is regular, then
BMOp,q = BMO1 (3.1)
in the sense of equivalent norm.
Proof. Now we consider 1 < p < \infty and 1 < q < \infty . As we known that Lp,q -space is a
rearrangement invariant Banach function space with lower and upper Boyd indices both equal to p
in this case. From the Theorem 3.4 of [19], one can obtain that
BMOp,q = BMO1, 1 < p, q < \infty . (3.2)
Combining Theorem 3.2 and (3.2), we have the formula (3.1).
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Received 15.04.16,
after revision — 09.12.17
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
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| id | umjimathkievua-article-1661 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:10:07Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-16612019-12-05T09:22:19Z A remark on John – Nirenberg theorem for martingales Зауваження щодо теореми Джона– Нiренберга для мартингалiв Li, L. Лі, Л. This paper is mainly devoted to establishing an extension of the John – Nirenberg theorem for martingales, more precisely, let $1 < p < \infty$ and $0 < q < \infty$. If the stochastic basis $(\scr {F_n})_n\geq 0$ is regular, then $BMO_{p,q} = BMO_1$ with the equivalent norms. Our method is to use a new atomic decomposition construction of the martingale Hardy space. Роботу, в основному, присвячено доведенню узагальнення теореми Джона – Нiренберга для мартингалiв, бiльш точно, для $1 < p < \infty$ та $0 < q < \infty$. За умови, що стохастичний базис $(\scr {F_n})_n\geq 0$ є регулярним, маємо $BMO_{p,q} = BMO_1$ з еквiвалентними нормами. Наш метод зводиться до застосування нової конструкцiї атомного розкладу простору мартингалiв Гардi. Institute of Mathematics, NAS of Ukraine 2018-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1661 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 11 (2018); 1571-1577 Український математичний журнал; Том 70 № 11 (2018); 1571-1577 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1661/643 Copyright (c) 2018 Li L. |
| spellingShingle | Li, L. Лі, Л. A remark on John – Nirenberg theorem for martingales |
| title | A remark on John – Nirenberg theorem for martingales |
| title_alt | Зауваження щодо теореми Джона– Нiренберга
для мартингалiв |
| title_full | A remark on John – Nirenberg theorem for martingales |
| title_fullStr | A remark on John – Nirenberg theorem for martingales |
| title_full_unstemmed | A remark on John – Nirenberg theorem for martingales |
| title_short | A remark on John – Nirenberg theorem for martingales |
| title_sort | remark on john – nirenberg theorem for martingales |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1661 |
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