# Atomic Decompositions and John-Nirenberg Theorem of Grand Martingale Hardy Spaces with Variable Exponents

Authors: Libo Li; Zhiwei Hao
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Letθ≥0 and p· be a variable exponent, and we introduce a new class of function spaces Lp·,θ in a probabilistic setting which unifies and generalizes the variable Lebesgue spaces with θ=0 and grand Lebesgue spaces with p·≡p and θ=1. Based on the new spaces, we introduce a kind of Hardy-type spaces, grand martingale Hardy spaces with variable exponents, via the martingale operators. The atomic decompositions and John-Nirenberg theorem shall be discussed in these new Hardy spaces.


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## Body

## 1. Introduction

The martingale theory is widely studied in the field of mathematical physics, stochastic analysis, and probability. Weisz [1] presented the atomic decomposition theorem for martingale Hardy spaces. Herz [2] established the John-Nirenberg theorem for martingales. Since then, the study of martingale Hardy spaces associated with various functional spaces has attracted a steadily increasing interest. For instance, martingale Orlicz-type Hardy spaces were investigated in [3–6], martingale Lorentz Hardy spaces were studied in [7–9], and variable martingale Hardy spaces were developed in [10–14].Let1<p<∞, and the grand Lebesgue space Lp)E introduced by Iwaniec and Sbordone [15] is defined as the Banach function space of the measurable functions f on finite E such that
(1)fLp=sup0<η<p−1η1∣E∣∫Efxp−ηdx1/p−η<∞.Such spaces can be used to integrate the Jacobian under minimal hypotheses [15]. The grand Lebesgue spaces as a kind of Banach function space were investigated in the papers of Capone et al. [16, 17], Fiorenza et al. [18–21], Kokilashvili et al. [22, 23], and so forth. In particular, grand Lebesgue spaces with variable exponents were studied in [24, 25].We find that the framework of grand Lebesgue spaces with variable exponents has not yet been studied in martingale theory. This paper is aimed at discussing the variable grand Hardy spaces defined on the probabilistic setting and showing the atomic decompositions and John-Nirenberg theorem in these new Hardy spaces. More precisely, we first present the atomic characterization of grand Hardy martingale spaces with variable exponents. To do so, we introduce the following new notations of atom.Definition 1.
Letp· be a variable exponent and θ≥0. A measurable function a is called a 1,p·,θ-atom (resp. 2,p·,θ-atom, 3,p·,θ-atom) if there exists a stopping time τ such that
a1Ena=0, ∀n≤τ
a2∥sa∥L∞resp.∥Sa∥L∞,MaL∞≤χτ<∞Lp·,θ−1.See Section2 for the notation Lp·,θ. Denote by Asp·,∞ (resp. ASp·,∞, AMp·,∞) the collection of all sequences of triplet ak,τk,μk, where ak are 1,p·,θ-atoms (resp. 2,p·,θ-atoms, 3,p·,θ-atoms), τk are stopping times satisfying a1 and a2 in Definition 1, and μk are nonnegative numbers and also
(2)∑k∈ℤμkχτk<∞∥χτk<∞∥Lp·,θLp·,θ<∞.Under these definitions, we show the atomic decompositions of the grand Hardy martingale spaces with variable exponents (see Section3). To be precise, we prove that for any f=fnn≥0, f∈Hp·,θs (resp. Qp·,θ, Dp·,θ) iff there exists a sequence of triplet ak,τk,μk∈Asp·,∞ (resp. ASp·,∞, AMp·,∞) so that for each n≥0,
(3)fn=∑k∈ℤμkEnaka.e.,∑k∈ℤμkχτk<∞∥χτk<∞∥Lp·,θLp·,θ≈fHp·,θsresp.fQp·,θ,fDp·,θ.Moreover, we extend the classical John-Nirenberg theorem to the grand variable Hardy martingale spaces. To be precise, under suitable conditions, we present the following one:(4)fBMOp·,θ≈fBMO1.See Theorem11 for the details. This conclusion improves the recent results [12, 26], respectively.Throughout this paper,ℤ, ℕ, and ℂ denote the integer set, nonnegative integer set, and complex numbers set, respectively. We denote by C the absolute positive constant, which can vary from line to line. The symbol A≲B stands for the inequality A≤CB. If we write A≈B, then it stands for A≲B≲A.


## 2. Preliminaries

### 2.1. Grand Lebesgue Spaces with Variable Exponents

LetΩ,F,ℙ be a probability space. An F-measurable function p·:Ω⟶0,∞ which is called a variable exponent. For convenience, we denote
(5)p−≔essinfpω:ω∈Ω,p+≔esssuppω:ω∈Ω,p−B=essinfpω:ω∈Bandp+B=esssuppω:ω∈B.Denote byP=PΩ the collection of all variable exponents p· satisfying with 1<p−≤p+<∞. The variable Lebesgue space Lp·=Lp·Ω consists of all F-measurable functions f such that for some λ>0,
(6)ρfλ=∫Ω∣fw∣λpwdℙ<∞.This leads to a Banach function space under the Luxemburg-Nakano norm(7)fLp·≡infλ>0:ρfλ≤1.Based on this, we begin with the definition of the grand Lebesgue space with variable exponent.Definition 2.
Suppose thatp·∈P and θ≥0. We define the grand Lebesgue space with variable exponent Lp·,θ=Lp·,θΩ as the set of all F-measurable functions f satisfying
(8)fLp·,θ≔sup0<η<p−−1ηθ/p−−ηfLp·−η<∞.The Grand Lebesgue space with variable exponent can unify and generalize the various function spaces. To be precise, ifθ=0, Lp·,θ degenerates to the variable Lebesgue space Lp·. If θ=1 and p·≡p, Lp·,θ becomes the grand Lebesgue space Lp).


### 2.2. Martingale Grand Hardy Spaces via Variable Exponents

LetFnn≥0 be a nondecreasing sequence of sub-σ-algebras of F sets with F=σ⋃n≥0Fn. The expectation operator and the conditional expectation operator relative to Fn are denoted by E and En, respectively. A sequence f=fnn≥0 of random variables is said to be a martingale, if fn is Fn-measurable, E∣fn∣<∞, and Enfn+1=fn for every n≥0. Denote M to be the set of all martingales f=fnn≥0 with respect to Fnn≥0 such that f0=0. For f∈M, write its martingale difference by dnf=fn−fn−1n≥0,f−1=0. Define the maximal function, the square function, and the conditional square function of f, respectively, as follows:
(9)Mmf=supn≤mfn,Mf=supn≥0fn,Smf=∑n=0mdfn21/2,Sf=∑n=0∞dfn21/2,smf=∑n=0mEn−1dfn21/2,sf=∑n=0∞En−1dfn21/2.LetΓ be the set of all sequences λnn≥0 of nondecreasing, nonnegative, and adapted functions, and λ∞≔limn⟶∞λn. For f∈M, p·∈P, and θ≥0, denote
(10)ΓQp·,θf=λnn≥0∈Γ:Snf≤λn−1n≥1,λ∞∈Lp·,θ,ΓDp·,θf=λnn≥0∈Γ:∣fn∣≤λn−1n≥1,λ∞∈Lp·,θ.Now we introduce the grand martingale Hardy spaces associated with variable exponents as follows:(11)Hp·,θ∗=f∈M:Mf∈Lp·,θ,∥f∥Hp·,θ∗=∥Mf∥Lp·,θ,Hp·,θS=f∈M:Sf∈Lp·,θ,∥f∥Hp·,θS=∥Sf∥Lp·,θ,Hp·,θs=f∈M:sf∈Lp·,θ,∥f∥Hp·,θs=∥sf∥Lp·,θ,Qp·,θ=f∈M:∥f∥Qp·,θ<∞,∥f∥Qp·,θ=infλnn≥0∈ΓQp·,θf∥λ∞∥Lp·,θ,Dp·,θ=f∈M:∥f∥Dp·,θ<∞,∥f∥Dp·,θ=infλnn≥0∈ΓDp·,θf∥λ∞∥Lp·,θ.The boundedLp·,θ-martingale spaces
(12)Lp·,θ=f=fnn≥0:supn≥0∥fn∥Lp·,θ<∞,where
(13)∥f∥Lp·,θ=supn≥0∥fn∥Lp·,θ.Remark 3.
Ifθ=0, then we obtain the definitions of Hp·∗, Hp·S, Hp·s, Qp·, and Dp·, respectively (see [10, 12, 27]). If we consider the special case θ=1 and p·≡p with the notations above, we obtain the definitions of Hp)∗, Hp)S, Hp)s, Qp), and Dp), respectively (see [26]). In addition, if p·≡p and θ=0, we obtain the martingale Hardy spaces Hq∗, HqS, Hqs, Qq, and Dq, respectively (see [28]).Refer to [29, 30] for more information on martingale theory.


## 2.1. Grand Lebesgue Spaces with Variable Exponents

LetΩ,F,ℙ be a probability space. An F-measurable function p·:Ω⟶0,∞ which is called a variable exponent. For convenience, we denote
(5)p−≔essinfpω:ω∈Ω,p+≔esssuppω:ω∈Ω,p−B=essinfpω:ω∈Bandp+B=esssuppω:ω∈B.Denote byP=PΩ the collection of all variable exponents p· satisfying with 1<p−≤p+<∞. The variable Lebesgue space Lp·=Lp·Ω consists of all F-measurable functions f such that for some λ>0,
(6)ρfλ=∫Ω∣fw∣λpwdℙ<∞.This leads to a Banach function space under the Luxemburg-Nakano norm(7)fLp·≡infλ>0:ρfλ≤1.Based on this, we begin with the definition of the grand Lebesgue space with variable exponent.Definition 2.
Suppose thatp·∈P and θ≥0. We define the grand Lebesgue space with variable exponent Lp·,θ=Lp·,θΩ as the set of all F-measurable functions f satisfying
(8)fLp·,θ≔sup0<η<p−−1ηθ/p−−ηfLp·−η<∞.The Grand Lebesgue space with variable exponent can unify and generalize the various function spaces. To be precise, ifθ=0, Lp·,θ degenerates to the variable Lebesgue space Lp·. If θ=1 and p·≡p, Lp·,θ becomes the grand Lebesgue space Lp).


## 2.2. Martingale Grand Hardy Spaces via Variable Exponents

LetFnn≥0 be a nondecreasing sequence of sub-σ-algebras of F sets with F=σ⋃n≥0Fn. The expectation operator and the conditional expectation operator relative to Fn are denoted by E and En, respectively. A sequence f=fnn≥0 of random variables is said to be a martingale, if fn is Fn-measurable, E∣fn∣<∞, and Enfn+1=fn for every n≥0. Denote M to be the set of all martingales f=fnn≥0 with respect to Fnn≥0 such that f0=0. For f∈M, write its martingale difference by dnf=fn−fn−1n≥0,f−1=0. Define the maximal function, the square function, and the conditional square function of f, respectively, as follows:
(9)Mmf=supn≤mfn,Mf=supn≥0fn,Smf=∑n=0mdfn21/2,Sf=∑n=0∞dfn21/2,smf=∑n=0mEn−1dfn21/2,sf=∑n=0∞En−1dfn21/2.LetΓ be the set of all sequences λnn≥0 of nondecreasing, nonnegative, and adapted functions, and λ∞≔limn⟶∞λn. For f∈M, p·∈P, and θ≥0, denote
(10)ΓQp·,θf=λnn≥0∈Γ:Snf≤λn−1n≥1,λ∞∈Lp·,θ,ΓDp·,θf=λnn≥0∈Γ:∣fn∣≤λn−1n≥1,λ∞∈Lp·,θ.Now we introduce the grand martingale Hardy spaces associated with variable exponents as follows:(11)Hp·,θ∗=f∈M:Mf∈Lp·,θ,∥f∥Hp·,θ∗=∥Mf∥Lp·,θ,Hp·,θS=f∈M:Sf∈Lp·,θ,∥f∥Hp·,θS=∥Sf∥Lp·,θ,Hp·,θs=f∈M:sf∈Lp·,θ,∥f∥Hp·,θs=∥sf∥Lp·,θ,Qp·,θ=f∈M:∥f∥Qp·,θ<∞,∥f∥Qp·,θ=infλnn≥0∈ΓQp·,θf∥λ∞∥Lp·,θ,Dp·,θ=f∈M:∥f∥Dp·,θ<∞,∥f∥Dp·,θ=infλnn≥0∈ΓDp·,θf∥λ∞∥Lp·,θ.The boundedLp·,θ-martingale spaces
(12)Lp·,θ=f=fnn≥0:supn≥0∥fn∥Lp·,θ<∞,where
(13)∥f∥Lp·,θ=supn≥0∥fn∥Lp·,θ.Remark 3.
Ifθ=0, then we obtain the definitions of Hp·∗, Hp·S, Hp·s, Qp·, and Dp·, respectively (see [10, 12, 27]). If we consider the special case θ=1 and p·≡p with the notations above, we obtain the definitions of Hp)∗, Hp)S, Hp)s, Qp), and Dp), respectively (see [26]). In addition, if p·≡p and θ=0, we obtain the martingale Hardy spaces Hq∗, HqS, Hqs, Qq, and Dq, respectively (see [28]).Refer to [29, 30] for more information on martingale theory.


## 3. Atomic Characterization

The method of atomic characterization plays an useful tool in martingale theory (see for instance [1, 4, 6, 31–33]). We shall construct the atomic characterizations for grand Hardy martingale spaces with variable exponents in this section.Theorem 4.
Letp·∈P and θ≥0. If the martingale f∈Hp·,θs, then there exists a sequence of triplet ak,τk,μk∈Asp·,∞ so that for each n≥0,
(14)∑k∈ℤμkEnak=fn,a.e.,(15)∑k∈ℤμkχτk<∞∥χτk<∞∥Lp·,θLp·,θ≲fHp·,θs.
Conversely, if the martingalef has a decomposition of (14), then
(16)∥f∥Hp·,θs≲inf∑k∈ℤμkχτk<∞∥χτk<∞∥Lp·,θLp·,θ,where the infimum is taken over all the admissible representations of (14).Proof.
Letf∈Hp·,θs. Now consider the stopping time for each k∈ℤ:
(17)τk=infn∈ℕ:sn+1f>2k.
It is easy to see that the sequence of these stopping times is nondecreasing. For each stopping timeτ, denote fnτ=fn∧τ. It is easy to write that
(18)fn=∑k∈ℤfnτk+1−fnτk.
For eachk∈ℤ, let μk=3·2kχτk<∞Lp·,θ. If μk≠0, we set
(19)ank=fnτk+1−fnτkμk,n∈ℕ.
Ifμk=0, we set ank=0 for each n∈ℕ. For each fixed k∈ℤ, ankn≥0 is a martingale. Since sfτk=sτkf≤2k, we get
(20)sank≤sfτk+1+sfτkμk≤χτk<∞Lp·,θ−1.
We can easily check thatankn≥0 is a bounded L2-martingale. Hence, there exists an element ak∈L2 such that Enak=ank. If n≤τk, then ank=0, and sak≤χτk<∞Lp·,θ−1. Consequently, it implies that ak is really a 1,p·,θ-atom.
DenoteΛk≔τk<∞. Knowing that τk<∞=sf>2k and τk is nondecreasing for each k∈ℤ, we obtain Λk+1⊆Λk. Now, we point out that
(21)∑k∈ℤ3·2kχΛkx=2∑k∈ℤ3·2kχΛk\Λk+1x.
Indeed, for a fixedx0∈Ω, there is k0∈ℤ so that x0∈Λk0 and x0∈Λk0+1, then we have
(22)∑k∈ℤ3·2kχΛkx0=∑k=−∞k03·2kχΛkx0=∑k=−∞k03·2k=3·2k0+1,∑k∈ℤ3·2kχΛk\Λk+1x0=∑k=−∞k03·2kχΛk\Λk+1x0=3·2k0.
This means(23)∑k∈ℤμkχτk<∞∥χτk<∞∥Lp·,θLp·,θ=∑k∈ℤ3·2kχτk<∞Lp·,θ=6∑k∈ℤ2kχΛk\Λk+1Lp·,θ=6sup0<η<p−−1ηθ/p−−ηinfλ>0:∫Ω∑k∈ℤ2kχΛk\Λk+1xλpx−ηdℙ≤1=6sup0<η<p−−1ηθ/p−−ηinfλ>0:∑k∈ℤ∫Λk\Λk+12kλpx−ηdℙ≤1=6sup0<η<p−−1ηθ/p−−ηinfλ>0:∑k∈ℤ∫Λk\Λk+1sfλpx−ηdℙ≤1≤6sup0<η<p−−1ηθ/p−−ηinfλ>0:∫Ωsfλpx−ηdℙ≤1=6sup0<η<p−−1ηθ/p−−η∥sf∥Lp·−η=6∥f∥Hp·,θs.
For the converse part, according to the definition of1,p·,θ-atom, we easily conclude
(24)sak=sakχτk<∞≤sakL∞χτk<∞≤χτk<∞Lp·,θ−1χτk<∞,where ak is the 1,p·,θ-atom and τk is the stopping time associated with ak which, when combined with the subadditivity of the operator s, yields
(25)sf≤∑k∈ℤμksak≤∑k∈ℤμkχτk<∞∥χτk<∞∥Lp·,θ.
This implies(26)∥f∥Hp·,θs=∥sf∥Lp·,θ≤∑k∈ℤμkχτk<∞∥χτk<∞∥Lp·,θLp·,θ.
Taking over all the admissible representations of (14) for f, we obtain the desired result.
Next, we will characterizeQp·,θ and Dp·,θ by atoms, respectively. The proof is similar to the proof of Theorem 4. For the completeness of this paper, we provide some details.Theorem 5.
Supposep·∈P and θ≥0. If the martingale f=fnn≥0∈Qp·,θ (resp. Dp·,θ), then there exists a sequence of triplet ak,τk,μk∈ASp·,∞ (resp. AMp·,∞) so that for each n∈ℕ,
(27)fn=∑k∈ℤμkEnak,(28)∑k∈ℤμkχτk<∞∥χτk<∞∥Lp·,θLp·,θ≲∥f∥Qp·,θresp.∥f∥Dp·,θ.
Conversely, if the martingalef=fnn≥0 has admissible representation (27), then f∈Qp·,θ (resp. Dp·,θ) and
(29)∥f∥Qp·,θresp.fDp·,θ≲inf∑k∈ℤμkχτk<∞∥χτk<∞∥Lp·,θLp·,θ,where the infimum is taken over all the admissible representations of (27).Proof.
The proof follows the ideas in Theorem4, so we omit some details. Suppose f=fnn≥0∈Qp·,θ (resp. Dp·,θ). We define stopping times as follows:
(30)τk=infn∈ℕ:λn>2k,inf∅=∞,where λnn≥0 is an adapted, nondecreasing sequence such that almost everywhere ∣Snf∣≤λn−1 (resp.∣fn∣≤λn−1) and λ∞∈Lp·,θ.
Letakk∈ℤ and μkk∈ℤ be defined as in the proof of Theorem 4. And replace Λk=τk<∞=sf>2k by the Λk=τk<∞=λ∞>2k. Then, we obtain that fn=∑k∈ℤμkEnak and (28) still hold.
For the converse part, write(31)λn=∑k∈ℤμk∥Sak∥L∞χτk≤nresp.λn=∑k∈ℤμk∥Mak∥L∞χτk≤n.
Clearly,λnn≥0 is a nonnegative, nondecreasing, and adapted sequence with Sn+1f≤λn (resp.∣fn∣≤λn). Thus, we get
(32)∥f∥Qp·,θresp.fDp·,θ=λ∞Lp·,θ≤∑k∈ℤμkχτ<∞∥χτ<∞∥p·,θp·,θ.Taking over all the admissible representations of (27) for f, we obtain the desired result.Remark 6.
Supposep·∈P and θ≥0. We conclude that the sum ∑k=MNμkak in Theorem 4 converges to f in Hp·,θs as M⟶−∞, N⟶∞. Indeed, it follows by the subadditive of the operator s, we get, for any M,N∈ℤ with M<N,
(33)sf−∑k=MNμkak≤sf−fτN+1+sfτM.Moreover,sf−fτN+1 is decreasing and convergent to 0 (a.e.) as N⟶∞, and sfτM is decreasing and convergent to 0 (a.e.) as M⟶−∞. From this and the dominated convergence theorem in Lp·−ε for 0<ε<p−−1 (see [34], Theorem 2.62), it follows that
(34)∥f−∑k=MNμkak∥Hp·,θs≤∥sf−fτN+1+sfτM∥Lp·,θ≤sup0<η<p−−1ηθ/p−−η∥sf−fτN+1∥Lp·−η+sup0<η<p−−1ηθ/p−−η∥sfτM∥Lp·−η⟶0asM⟶−∞,N⟶∞.Furthermore, we can also show the norm convergence of the summation∑k=MNμkak in Theorems 5 as M⟶−∞, N⟶∞.


## 4. The Generalized John-Nirenberg Theorem

In the sequel of this section, we will often suppose that everyFn is generated by countably many atoms. Recall that B∈Fn is called an atom, and if for any A⊆B with A∈Fn satisfying ℙA<ℙB, we have ℙA=0. We denote by AFn the set of all atoms in Fn. We shall present the generalized John-Nirenberg theorem on grand Lebesgue spaces with variable exponents. For each 1≤p<∞, the Banach space BMOp (bounded mean oscillation [35]) is defined as
(35)BMOp=f∈Lp:fBMOp=supn≥1Enf−En−1fpL∞1/p<∞.It can be easily shown that the norm ofBMOp is equivalent to
(36)fBMOp=supτ∈T∥f−fτ−1∥Lp∥χτ<∞∥Lp,where T consists of all stopping times relative to Fnn≥0. It follows immediately from the John-Nirenberg theorem [2, 30] that
(37)BMOp=BMO1,1<p<∞.What is more, in [2], there has
(38)C·pfBMO1≥fBMOp≥fBMO1.Definition 7.
Forp·∈P and θ≥0, the generalized BMO martingale space is defined by
(39)BMOp·,θ=f∈Lp·,θ:∥f∥BMOp·,θ<∞,where
(40)∥f∥BMOp·,θ=supτ∈T∥f−fτ−1∥Lp·,θ∥χτ<∞∥Lp·,θ.Remark 8.
Ifθ=0, BMOp·,θ degenerates to the variable exponent BMO martingale space BMOp· introduced and studied in [12]. If θ=1 and p·≡p, BMOp·,θ becomes the grand BMO martingale space BMOp) studied in [26].In order to establish the generalized John-Nirenberg theorem in the framework ofBMOp·,θ, we need the following lemmas and notations.Lemma 9 (Hölder’s inequality, see [34]).
Letp·,q·,r·∈P satisfy
(41)1pω+1qω=1rω,a.e.ω∈Ω.
Then, there exists a constantC such that for all f∈Lp· and g∈Lq·, we have fg∈Lr· and
(42)∥fg∥Lr·≤CfLp·∥g∥Lq·.We mention that if the variable exponentpx meets the log-Hölder continuity condition in Euclidean spaces, the inverse Hölder’s inequality holds for characteristic functions in Lp·ℝn (see [36]). Compared with Euclidean space ℝn, the probability space Ω,ℙ has no natural metric structure. Fortunately, Jiao et al. [11, 27] put forward the following condition: there exists an absolute constant κ≥1 depending only on p· such that
(43)ℙBp−B−p+B≤κ,∀B∈⋃n≥0AFn.Lemma 10 (see [27]).
Supposep·∈P satisfying (43).
(1)
For eachB∈⋃n≥0AFn, we get(44)ℙB≈∥χB∥Lp·∥χB∥Lp′·.(2)
Letq·∈P satisfy (43). If r· satisfies(45)1rω=1pω+1qω,a.e.ω∈Ω,then r· also satisfies condition (43). Moreover, for each B∈⋃n≥0AFn, we deduce
(46)∥χB∥Lr·≈∥χB∥Lp·∥χB∥Lq·.Theorem 11.
Suppose thatp·∈P satisfies (43) and θ≥0. Then, for every f∈BMO1, there has
(47)fBMO1≲fBMOp·,θ≲fBMO1.Proof.
Ifp·∈P satisfies (43), then we clearly get that p·−η also satisfies (43) for 0<η<p−−1. It follows from Lemmas 9 and 10 that
(48)∥f−fτ−1∥L1χτ<∞L1≤∥f−fτ−1∥Lp·−η∥χτ<∞∥Lp·−η′χτ<∞L1≈∥f−fτ−1∥Lp·−ηχτ<∞Lp·−η,for any 0<η<p−−1. Here, the variable exponent p·−η′ is defined by
(49)1pω−η′+1pω−η=1,a.e.ω∈Ω.
This is equivalent to the following inequality:(50)∥f−fτ−1∥L1χτ<∞L1·χτ<∞Lp·−η≲∥f−fτ−1∥Lp·−η.
Hence, we have(51)∥f−fτ−1∥L1χτ<∞L1=sup0<η<p−−1ηθ/p−−η∥f−fτ−1∥L1/χτ<∞L1·χτ<∞Lp−ηsup0<η<p−−1ηθ/p−−η∥χτ<∞∥Lp·−η≲sup0<η<p−−1ηθ/p−−η∥f−fτ−1∥Lp·−ηsup0<η<p−−1ηθ/p−−η∥χτ<∞∥Lp·−η=∥f−fτ−1∥Lp·,θχτ<∞Lp·,θ.
Taking the supremum over all stopping times, we deduce(52)∥f∥BMO1≤∥f∥BMOp·,θ.
Conversely, from the definition ofLp·,θ, we get
(53)∥f−fτ−1∥Lp·,θχτ<∞Lp·,θ=sup0<η<p−−1ηθ/p−−η∥f−fτ−1∥Lp·−ηsup0<η<p−−1ηθ/p−−η∥χτ<∞∥Lp·−η≤sup0<η<p−−1ηθ/p−−η∥f−fτ−1∥Lp·−ηηθ/p−−η∥χτ<∞∥Lp·−η=sup0<η<p−−1∥f−fτ−1∥Lp·−η∥χτ<∞∥Lp·−η.
It follows from Lemma9 that
(54)∥f−fτ−1∥Lp·−η∥χτ<∞∥Lp·−η≤∥f−fτ−1∥L2p+∥χτ<∞∥Lq·∥χτ<∞∥Lp·−η≈∥f−fτ−1∥L2p+∥χτ<∞∥L2p+,where q· satisfies
(55)1pω−η=12p++1qω,a.e.ω∈Ω.
Hence, by (38), we deduce that
(56)∥f∥BMOp·,θ=supτ∈T∥f−fτ−1∥Lp·,θχτ<∞Lp·,θ≲supτ∈Tsup0<η<p−−1∥f−fτ−1∥L2p+∥χτ<∞∥L2p+=supτ∈T∥f−fτ−1∥L2p+∥χτ<∞∥L2p+=∥f∥BMO2p+≤C·2p+∥f∥BMO1.
From what has been discussed above, we draw the conclusion that(57)∥f∥BMO1≲∥f∥BMOp·,θ≲∥f∥BMO1.
Theorem11 improves the recent results [12, 26], respectively. More precisely, if we consider the case θ=0, then the following result holds:Corollary 12.
Ifp· satisfies (43) with 1<p−≤p+<∞, then for f∈BMO1,
(58)fBMOp·≈fBMO1.And especially forθ=1 and p·≡p, we get the conclusion as follows.Corollary 13 (see [26]).
Suppose1<p<∞, then for f∈BMO1,
(59)fBMOp≈fBMO1.



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*Source: 9021391-2022-01-19.xml*