# The Boundedness of Doob’s Maximal and Fractional Integral Operators for Generalized Grand Morrey-Martingale Spaces

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

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## Abstract

In this paper, we introduce the generalized grand Morrey spaces in the framework of probability space setting in the spirit of the martingale theory and grand Morrey spaces. The Doob maximal inequalities on the generalized grand Morrey spaces are provided. Moreover, we present the boundedness of fractional integral operators for regular martingales in this new framework.


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## Body

## 1. Introduction

A real-valued functionf is said to belong to the Morrey space Lp,λℝN on the N-dimensional Euclidean space ℝN provided the following norm is finite:
(1)fLp,λℝN=supx,r∈ℝN×ℝ+rλ−N∫Bx,rfypdy1/p.Here1≤p<∞, 0≤λ≤N, ℝ+=0,∞, and Bx,r are a ball in ℝN centered at x of radius r. This class of functions was first introduced by Morrey [1] in order to study regularity problem arising in Calculus of Variations, describe local regularity more precisely than Lebesgue spaces. In the past, Morrey spaces have been studied heavily, such as the maximal operators, fractional integral operators, and singular operators. The results are extensively applied not only in partial differential equations but also in harmonic analysis. We refer the readers to [2, 3] and the references therein.The Morrey spaces on Euclidean spaces have been developed to the generalization versions, for example, the generalized Morrey spaces [4, 5], the Orlicz-Morrey spaces [6, 7], the Triebel-Lizorkin-Morrey spaces [8], and the variable exponent Morrey spaces [9]. Especially, Meskhi [10] introduced the grand Morrey spaces and established the boundedness of the Hardy-Littlewood maximal, Calderón-Zygmund, and potential operators in these spaces. The generalized grand Morrey spaces in a general setting of the quasi-metric measure spaces are studied by Kokilashvili et al. [11, 12].Moreover, in probability theory, Nakai and Sadasue [13] introduced Morrey spaces of martingales as the following:LetΩ,F,ℙ be a probability space and Fnn≥0 be a nondecreasing sequence of sub-σ-algebras of F such that F=σ∪n≥0Fn.We assume that everyσ-algebra Fn is generated by countable atoms, where B∈Fn is called an atom, if any A⊂B with A∈Fn satisfies ℙA=0 or ℙA=ℙB. Denote by AFn the set of all atoms in Fn. For p∈1,∞ and μ∈−∞,∞, martingale Morrey space Lp,μΩ consists of all f∈L1Ω having the finite norm
(2)fLp,μΩ=supn≥0supB∈AFn1ℙBμ1ℙB∫Bfpdℙ1/p.They introduced some basic properties of the martingale Morrey spaces. Furthermore, the Doob maximal inequality was established, and the mapping properties for the fractional integral operators were investigated on these spaces. Two generalized versions of them introduced in [14, 15]. Ho [16] presented atomic decompositions of martingale Hardy-Morrey spaces. Later on, he [17] introduced a version of martingale Morrey spaces equipping with Banach function spaces. Jiao et al. [18] studied the maximal operator, atom decompositions, and fractional integral operators on martingale Morrey spaces with variable exponents.Recently, Deng and Li [19] studied the Doob maximal operator and fractional integral operator in the framework of grand Morrey-martingale spaces associated with an almost decreasing function. Moreover, compared with classical martingale spaces, the grand martingale spaces have not of absolutely continuous norm based on [20]. Consequently, we need a further research about grand martingale spaces. Motivated by the works of this and [11], the paper is to investigate the generalized grand Morrey space theory for the martingale setting. More precisely, we first introduce the generalized grand Morrey-martingale spaces and then establish the Doob maximal inequality in this new framework. As an application, we discuss the boundedness of fractional integral operators for regular martingales in the generalized grand Morrey-martingale spaces.


## 2. Preliminaries

Now we recall some standard notations from martingale theory. Refer to [21, 22] for more information on martingale theory. The expectation is denoted by E with respect to Ω,F,ℙ. Recall that the conditional expectation operator relative to Fn is denoted by En, i.e., EfFn=Enf. A sequence of measurable functions f=fnn≥0⊂L1Ω is called a martingale with respect to Fnn≥0 if Enfn+1=fn for every n≥0. Let M be the set of all martingale f=fnn≥0 relative to Fnn≥0 such that f0=0. For f∈M, denote its martingale difference by dnf=fn−fn−1 (n≥0, with convention d0f=0).The maximal function off∈M is defined by
(3)Mmf=supn≤mfn,Mf=supn≥0fn.Forp>1 and f∈LpM, we have
(4)MfLp≤pp−1fLp,which is well known in the literature as the Doob maximal inequality (see [22]).Hence, it follows from the above inequality that ifp∈1,∞, then Lp-bounded martingale converges in Lp. Moreover, if p∈1,∞, then, for any f∈Lp, its corresponding martingale fnn≥0 with fn=Enf is an Lp-bounded martingale and converges to f in Lp (see [21]). For this reason, a function f∈L1 and the corresponding martingale fnn≥0 will be denoted by the same symbol f.It is convenient for us to state the generalized grand Morrey-martingale spaces, we first need to recall the definition of martingale Morrey spacesLp,λ=Lp,λΩ as follows.Definition 1.
Forp∈1,∞ and λ∈−∞,∞, let
(5)Lp,λ=f∈M:fLp,λ<∞,where
(6)fLp,λ=supn≥0supB∈AFn1ℙBλ∫Bfpdℙ1/p.Remark 2.
Ifλ=pu+1, the above definition of ·Lp,λ is equivalent to ·Lp,μ (see (2)), which introduced by Nakai and Sadasue [13].
Ifλ=0 and F0=∅,Ω, then the Morrey-martingale space Lp,λ is Lp by the above definition.Now we introduce a new type Morrey-martingale spaces as follows.Definition 3.
Let1<p<∞, 0≤λ<1, φ be a nondecreasing real-valued nonnegative function defined on 0,p−1 with limx⟶0+φx=0, and δ be a positive number. The generalized grand Morrey-martingale space Lp,λδ,φΩ consists of f∈M such that
(7)fLp,λδ,φ=sup0<ε≤ss=minp−1,αεδ/p−εsupn≥0supB∈AFn1ℙBλ−φε∫Bfp−εdℙ1/p−εis finite, where α=supx>0:φx≤λ.Notice that, in the above condition,·Lpλδ,φ is a norm and can be expressed as
(8)fLp,λδ,φ=sup0<ε≤ss=minp−1,αεδ/p−εfLp−ε,λ−φε.Remark 4.
Ifλ>0 and φ≡0, then Lp,λδ,φΩ is called grand Morrey-martingale space, which was introduced in [19]. If λ=0, δ=1, and φ≡0, we recover the grand Lebesgue spaces for martingales introduced in [23]. In this case, if consider Ω=0,1, we have grand Lebesgue spaces introduced in [24]. We mention that there exists a martingale f=fnn≥0 such that it does not converge in Lp0,1. Indeed, Lp0,1 is a rearrangement-invariant Banach function space, Lp0,1≠L10,1, but is not of absolutely continuous norm from [20]. According to Theorem 3.3 in [25], there exists a martingale f=fnn≥0 such that it does not converge in Lp0,1.The stochastic basisFnn≥0 is said to be regular, if there exists a constant R≥2 such that
(9)fn≤Rfn−1holds for all nonnegative martingale fnn≥0 adapted to Fnn≥0.For regular stochastic basis, there has the following property, proved in [13].Lemma 5.
LetFnn≥0 be regular. Then, for every sequence
(10)Bn⊂Bn−1⊂⋯⊂Bk⊂⋯⊂B0,Bk∈AFk,we have
(11)Bk=Bk−1or1+1RℙBk≤ℙBk−1≤RℙBk1≤k≤n,where R is the positive constant in (9).


## 3. The Doob Maximal Operator

In this section, we present the boundedness of Doob’s maximal operator on generalized grand Morrey-martingale spaces.Theorem 6.
Let1<p<∞, δ>0, and 0≤λ<1. Then,
(12)MfLp,λδ,φ≤CfLp,λδ,φ,where the constant C satisfies
(13)C=inf0<θ<ssδ/p−εθ−δ/p−θp−θp−θ−1+1,which only depends on the parameters p,λ,δ,φ for s=minp−1,α and α=supx>0:φx≤λ.In order to prove Theorem6, we need the following useful lemma:Lemma 7.
Letf=fnn≥0∈L1,1<p<∞,0≤λ<1. Then,
(14)MfLp,λ≤pp−1+1fLp,λ.Proof.
For anyB∈AFm and m≥0, suppose that f=g+h and g=fχB.
Then, according to the well-known Doob’s maximal inequality, that is,(15)MfLp≤pp−1fLp,we have
(16)∫BMgpdℙ≤∫ΩMgpdℙ≤pp−1p∫Ωgpdℙ=pp−1p∫Bfpdℙ.
Hence,(17)1ℙBλ∫BMgpdℙ1/p≤pp−1fLp,λ.
Next, takeBn∈AFn,n=0,1,⋯,m, such that B=Bm⊂Bm−1⊂⋯⊂B0. Then, for a.e. ω∈B,
(18)Enhω=0,ifn≥m,1ℙBn∫Bnhdℙ,ifn<m.
Ifn<m, according to Jensen’s inequality, then
(19)Enhω≤1ℙBn∫Bnhpdℙ1/p≤ℙBnλ−1/pfLp,λ≤ℙBλ−1/pfLp,λ,
where the last inequality dues to0≤λ<1 and ℙB≤ℙBn. This means
(20)Mhω≤ℙBλ−1/pfLp,λforanyω∈B.
Then, we obtain(21)1ℙBλ∫BMhpdℙ1/p≤fLp,λ.
Combining inequalities (17) and (21) and Mf≤Mg+Mh, we can get
(22)1ℙBλ∫BMfpdℙ1/p≤pp−1+1fLp,λ.
The proof is complete.Note that Nakai and Sadasue [13] proved that, for 1<p<∞ and −1/p≤μ<0,
(23)MfLp,μ≤CpfLp,μ.The proof of Lemma7 is devoted to determination of the constant Cp. Now we prove Theorem 6:Proof.
Let0<θ<s, and we have
(24)MfLp,λδ,φ=sup0<ε≤ss=minp−1,αεδ/p−εMfLp−ε,λ−φε=maxsup0<ε<θεδ/p−εMfLp−ε,λ−φε,supθ≤ε<ss=minp−1,αεδ/p−εMfLp−ε,λ−φε,where α=supx>0:φx≤λ. Let
(25)I=supθ≤ε<ss=minp−1,αεδ/p−εMfLp−ε,λ−φε.
Note that the functionhε≔εδ/p−ε is increasing in 0<ε<p, which means
(26)I≤sδ/p−ssupθ≤ε<ss=minp−1,αsupn≥0supB∈AFn1ℙBλ−φε∫BMfp−εdℙ1/p−ε≤sδ/p−ssupθ≤ε<ss=minp−1,αsupn≥0supB∈AFn1ℙBλ−φε−1/p−ε1ℙB∫BMfp−εdℙ1/p−ε≤sδ/p−ssupθ≤ε<ss=minp−1,αsupn≥0supB∈AFnθδ/p−θθ−δ/p−θℙBΔε,θ1ℙBλ−φθ∫BMfp−θdℙ1/p−θ,where
(27)Δε,θ≔λ−φε−1p−ε−λ−φθ−1p−θ.
Note that forθ≤ε,
(28)Δε,θ=ε−θλ−1+φθp−ε−φεp−θp−εp−θ≤ε−θλ−1+φεp−ε−φεp−θp−εp−θ≤ε−θλ−1+φεθ−εp−εp−θ≤0.
Then,0<1/ℙBΔε,θ≤1, and we obtain I≤sδ/p−sθ−δ/p−θθδ/p−θMfLp−θ,λ−φθ. Obviously, sδ/p−sθ−δ/p−θ>1 as 0<θ<s. Thus, according to Lemma 7, we deduce that
(29)MfLp,λδ,φ≤sδ/p−sθ−δ/p−θsup0<ε≤θεδ/p−εMfLp−ε,λ−φε≤sδ/p−sθ−δ/p−θsup0<ε≤θεδ/p−εp−εp−ε−1+1fLp−ε,λ−φε≤sδ/p−sθ−δ/p−θp−θp−θ−1+1fLp,λδ,φ.
Taking the infimum over allθ, we obtain that
(30)MfLp,λδ,φ≤CfLp,λδ,φ, where
(31)C=inf0<θ<ss=minp−1,αα=supx>0:φx≤λsδ/p−sθ−δ/p−θp−θp−θ−1+1∈1,∞.


## 4. The Fractional Integral Operator

In this section, we present the boundedness of the fractional integral operator in the new type grand Morrey-martingale spaces. In martingale theory, Chao and Ombe [26] introduced the fractional integrals for dyadic martingales. The fractional integrals in this section are defined for more general martingale setting as in [13, 14] (see also [15, 27–32]).Definition 8.
Letf=fnn≥0∈M and ι>0, and the fractional integral Iιf=Iιfnn≥0 of martingale f is defined by
(32)Iιfn=∑k=0nbk−1ιdkf,where bk is an Fk-measurable function such that
(33)bkω=∑B∈AFkℙBχBω,ω∈Ω.Remark 9.
Obviously,bk is bounded in above definition; there Iιf=Iιfnn≥0 is a martingale transform of f.The following lemma was shown in [13]. Here we focus on more accurate upper boundedness.Lemma 10.
Suppose thatFnn≥0 is regular. Let 1<p<∞, 1≤q≤v/up, −1/p≤v<0, and u=v+ι<0. Then, for f∈L1,
(34)MIιfLq,u≤Cq,u,p,vfLp,v,where Cq,u,p,v=1+1/Rv/1−1+1/Ru+2p/p−1+1p/q and R is the constant in formula (9).Proof.
SincefLq1,λ≤fLq2,λ for q1≤q2 by Hölder’s inequality, it is enough to prove it in the case where q=v/up. Without loss of generality, we let fLp,v≠0.
First, we prove the following inequality holds for anyn≥1, and any Bn∈AFn,
(35)Iιfnω≤1+1/Rv1−1+1/Ru+2Mfωu/vfLp,v−ι/v,ω∈Bn.
ChooseBk∈AFk and 0≤k<n, such that Bn⊂Bn−1⊂⋯⊂B0, and let
(36)K=k:0<k≤n,Bk≠Bk−1=k1,k2,⋯,kh,where 0=k0<k1<k2<⋯<kh.
SinceFnn≥0 is regular, according to Lemma 5, we have
(37)1+1Rbkj≤bkj−1≤RbkjonBn.
So, fork∉K, we have bk=bk−1 and dkf=0. Hence, we obtain
(38)Iιfn=∑0<kj≤nbkj−1ιdkjf=∑j=1hbkj−1ιdkjfonBn.
Forω∈Bn,
(39)dkjfω=fkjω−fkj−1ω≤fkjω+fkj−1ω=1ℙBkj∫Bkjfdℙ+1ℙBkj−1∫Bkj−1fdℙ≤1ℙBkj∫Bkjfpdℙ1/p+1ℙBkj−1∫Bkj−1fpdℙ1/p≤ℙBkjv+ℙBkj−1vfLp,v≤1+1Rvbkj−1ωvfLp,v.
Then, forω∈Bn and when 0<k≤m where m≤n,
(40)∑k=0mbk−1ωιdkfω≤1+1Rv∑0<kj≤mbkj−1ωv+ιfLp,v=1+1Rv∑0<kj≤mbkj−1ωufLp,v≤1+1/Rv1−1+1/RubmωufLp,v.
Forω∈Bn and when m+1≤k<n, let jk=minj:k<kj, and we have
(41)∑k=m+1nbk−1ωιdkfω=∑j=jkhbkj−1ωιdkjfω=∑j=jkhbkj−1ωιfkjω−∑j=jkhbkj−1ωιfkj−1ω=bkh−1ωιfkhω+∑j=jkh−1bkj−1ωι−bkjωιfkjω−bkjk−1ωιfkjk−1ω≤bkh−1ωιMfω+∑j=jkh−1bkj−1ωι−bkjωιMfω+bkjk−1ωιMfω≤2bkjk−1ωιMfω=2bmωιMfω.
Now let(42)Λ1=ω∈Ω:MfωfLp,v1/v≤b0ωandΛ2=Ω\Λ1.
Next we estimateIιfn from the following cases. For the first case, if ω∈Λ1∩Bn and
(43)MfωfLp,v1/v≤bnω,then, by formula (40) and u=v+ι<0, we have
(44)Iιfnω≤1+1/Rv1−1+1/RubnωufLp,v≤1+1/Rv1−1+1/RuMfωfLp,vu/vfLp,v=1+1/Rv1−1+1/RuMfωu/vfLp,v−ι/v.
For the second case, ifω∈Λ1∩Bn and
(45)bnω<MfωfLp,v1/v,then there exists m such that
(46)1Rbmω<MfωfLp,v1/v≤bmω.
Combining (46) with formulas (40) and (41), we have
(47)Iιfnω≤1+1/Rv1−1+1/RubmωufLp,v+2bmωιMfω≤1+1/Rv1−1+1/RuMfωfLp,vu/vfLp,v+2MfωfLp,vι/vMfω≤1+1/Rv1−1+1/Ru+2Mfωu/vfLp,v−ι/v.
For the third case, ifω∈Λ2∩Bn, then by (41), we have
(48)Iιfnω≤2b0ωιMfω≤2MfωfLp,vι/vMfω=2Mfωu/vfLp,v−ι/v.
Formulas (44), (47), and (48) give that
(49)Iιfnω≤1+1/Rv1−1+1/Ru+2Mfωu/vfLp,v−ι/v,ω∈Bn,which implies that
(50)1ℙB∫BMIιfωqdℙ1/q≤1+1/Rv1−1+1/Ru+21ℙB∫BMfωqu/vdℙ1/qfLp,v−ι/v=1+1/Rv1−1+1/Ru+21ℙB∫BMfωpdℙ1/pp/qfLp,v1−p/q.
Moreover, by Lemma7, we have
(51)1ℙB∫BMfωpdℙ1/pp/q≤ℙBvMfLp,vp/q≤pp−1+1p/qℙBufLp,vp/q.
It follows from the above inequality and (50) that
(52)1ℙB∫BMIιfωqdℙ1/q≤1+1/Rv1−1+1/Ru+2pp−1+1p/qℙBufLp,v,that is to say,
(53)MIιfLq,u≤1+1/Rv1−1+1/Ru+2pp−1+1p/qfLp,v.
The proof is complete.Theorem 11.
Let1<q<∞, 0≤λ<1, 0<ι<1−λ/q, 1/q−1/p=ι/1−λ, δ2>0, and δ1≥δ21+ιp/1−λ. Suppose that φ1 and φ2 are continuous nonnegative and nondecreasing real-valued functions on 0,p−1 and 0,q−1, respectively, satisfying
(i)
φ1∈C10,κ for some positive κ>0(ii)
limx⟶0+φ1x=0(iii)
0≤limx⟶0+dφ1x/dx<1−λ2/ιp2(iv)
φ2η=φ1ϕ−1η, where ϕ−1 is the inverse of ϕ on 0,κ for κ>0, and ϕx=q−p−x1−λ+φ1x/1−λ+φ1x+ιp−x. Then, for f∈Lq,λδ2,φ2,(54)MIιfLp,λδ1,φ1≤Cp,δ1,δ2,φ1,λfLq,λδ2,φ2,where Cp,δ1,δ2,φ1,λ only depends on p, δ1, δ2, φ1, and λ.Proof.
The equation1/q−1/p=ι/1−λ and limx⟶0+φ1x=0 give that limx⟶0+ϕx=0. The condition (iii) ensures that limx⟶0+dϕx/dx>0. Then, there exists small positive number ϵ<min1,κ such that ϕ is increasing in 0,ϵ and ϕϵ<q−1/2. Now fix
(55)θ∈0,mins,ϵ,where s=minp−1,α and α=supx>0:φ1x≤λ.
Firstly, we consider the case ofε∈θ,s. In this situation, let
(56)Iε≔εδ1/p−ε1ℙBλ−φ1ε∫BMIιfp−εdℙ1/p−ε.
Sincep−ε<p−θ, then it follows from Jensen’s inequality that
(57)Iε=εδ1/p−εℙBφ1ε+1−λ/p−ε1ℙB∫BMIιfp−εdℙ1/p−ε≤εδ1/p−εℙBφ1ε+1−λ/p−ε1ℙB∫BMIιfp−θdℙ1/p−θ.
Note thatφ1x+1−λ/p−x is a nonnegative and nondecreasing function on θ,s; hence,
(58)Iε≤εδ1/p−εℙBφ1θ+1−λ/p−θ1ℙB∫BMIιfp−θdℙ1/p−θ≤sδ1/p−sθ−δ1/p−θsup0<t≤θtδ1/p−tsupn≥0supB∈AFn1ℙBλ−φ1t∫BMIιfp−tdℙ1/p−t.
Sincesδ1/p−sθ−δ1/p−θ>1 for θ<s, then the following inequality holds
(59)MIιfLp,λδ1,φ1≤sδ1/p−sθ−δ1/p−θsup0<t≤θtδ1/p−tsupn≥0supB∈AFn1ℙBλ−φ1t∫BMIιfp−tdℙ1/p−t.
Next, we considert∈0,θ in the following discussion. Since 1/q−1/p=ι/1−λ, we can choose η and t satisfying
(60)1q−η−1p−t=ι1−λ+φ1t.
Obviously we know thatt⟶0 if and only if η⟶0, and we obtain η with respect to t as follows:
(61)η=q−p−t1−λ+φ1t1−λ+φ1t+ιp−t=ϕt.
Let(62)p~=q−η,q~=p−t,v~=λ−φ1t−1q−η,u~=λ−φ1t−1p−t.
It is not hard to see that1≤q~=v~/u~p~, −1/p~≤v~<0, v~+ι=u~≤−1−λ/p−t<0, and
(63)q−η=q−ϕt≥q−ϕϵ≥q+12>1.
Moreover,(64)Cq~,u~,p~,v~=1+1/Rv~1−1+1/Ru~+2p~p~−1+1p~/q~≤1+R1+φ1θ1−R/R+11−λ/p+2q−ϕεq−ϕε−1+1q≤1+R1+φ1θ1−R/R+11−λ/p+2q−ϕθq−ϕθ−1+1q≕Cθ.
Notice thatφ1θ≤φ1s and q−ϕθ−1≥q−ϕϵ−1≥q−1/2. This implies that Cθ<∞.
Thus, according to the inequalities (59) and (64) and Lemma 10, we have
(65)MIιfLp,λδ1,φ1≤sδ1/p−sθ−δ1/p−θsup0<t≤θtδ1/p−tsupn≥0supB∈AFn1ℙBλ−φ1t−1/p−t1ℙB∫BMIιfp−tdℙ1/p−t=sδ1/p−sθ−δ1/p−θsup0<t≤θtδ1/p−tMIιfLq~,u~≤Cθsδ1/p−sθ−δ1/p−θsup0<t≤θtδ1/p−tfLp~,v~=Cθsδ1/p−sθ−δ1/p−θsup0<t≤θtδ1/p−tsupn≥0supB∈AFn1ℙBλ−φ1t−1/q−η1ℙB∫Bfq−ηdℙ1/q−η=Cθsδ1/p−sθ−δ1/p−θsup0<t≤θtδ1/p−tsupn≥0supB∈AFn1ℙBλ−φ1t∫Bfq−ηdℙ1/q−η≤Cθsδ1/p−sθ−δ1/p−θsup0<t≤θtδ1/p−tη−δ2/q−ηfLq,λδ2,φ2,where the last inequality holds because of φ2η=φ1ϕ−1η=φ1t.
Finally, we shall show thatsup0<t≤θtδ1/p−tη−δ2/q−η is bounded. Since limx⟶0+ϕx=0, by l’Hospital’s rule, we have
(66)limx⟶0+ϕxx=limx⟶0+ϕ′xx′=limx⟶0+1−λ2−ιp2φ1′x1−λ+ιp2.
Combining with the condition (iii), we haveϕx~x as x⟶0+. This implies
(67)η−δ2/q−η~t−δ2/q−η,ast⟶0+.
Moreover, usingδ1≥δ21+ιp/1−λ, 0<t≤θ<1, and formula (60), we obtain
(68)tδ1/p−tη−δ2/q−η≤Ctδ21+ιp/1−λ/p−tt−δ21/p−t+ι/1−λ+φ1t=Ctιδ2p/1−λp−t−1/1−λ+φ1t.
Obviously,p/1−λp−t−1/1−λ+φ1t>0, which implies that sup0<t≤θtδ1/p−tη−δ2/q−η≤C is bounded.
To sum up, we have(69)MIιfLp,λδ1,φ1≤Cp,δ1,δ2,φ1,λfLq,λδ2,φ2,where
(70)Cp,δ1,δ2,φ1,λ=Csδ1/p−sinfθ∈0,mins,ϵCθθ−δ1/p−θ.Remark 12.
Recently, new results concerning the grand variable exponent Lebesgue spaces for martingales have emerged (see [33]).



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