# Martingale Morrey-Campanato Spaces and Fractional Integrals

Authors: Eiichi Nakai; Gaku Sadasue
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

We introduce Morrey-Campanato spaces of martingales and give their basic properties. Our definition of martingale Morrey-Campanato spaces is different from martingale Lipschitz spaces introduced by Weisz, while Campanato spaces contain Lipschitz spaces as special cases. We also give the relation between these definitions. Moreover, we establish the boundedness of fractional integrals as martingale transforms on these spaces. To do this we show the boundedness of the maximal function on martingale Morrey-Campanato spaces.


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## Body

## 1. Introduction

The purpose of this paper is to introduce Morrey-Campanato spaces of martingales. The Lebesgue spaceLp plays an important role in martingale theory as well as in harmonic analysis. Moreover, in martingale theory, Lorentz spaces, Orlicz spaces, Hardy spaces, Lipschitz spaces, and John-Nirenberg space BMO also have been developed by many authors, see [1–5], and so forth. Recently Kikuchi [6] investigated Banach function spaces of martingales. In this paper we introduce Morrey-Campanato spaces of martingales and give their basic properties. Moreover, we establish the boundedness of fractional integrals as martingale transforms on these spaces. Note that Campanato spaces are not Banach function spaces in general.We consider a probability space(Ω,ℱ,P) such that ℱ=σ(⋃nℱn), where {ℱn}n≥0 is a nondecreasing sequence of sub-σ-algebras of ℱ. Following Weisz [5], we call {ℱn}n≥0 a stochastic basis. For the sake of simplicity, let ℱ-1=ℱ0. We suppose that every σ-algebra ℱn is generated by countable atoms, where B∈ℱn is called an atom (more precisely a (ℱn,P)-atom), if any A⊂B with A∈ℱn satisfies P(A)=P(B) or P(A)=0. Denote by A(ℱn) the set of all atoms in ℱn. The expectation operator and the conditional expectation operators relative to ℱn are denoted by E and En, respectively.We define Morrey-Campanato spaces as the following: letp∈[1,∞) and λ∈(-∞,∞). For f∈L1, let(1.1)‖f‖Lp,λ=sup⁡n≥0sup⁡B∈A(Fn)1P(B)λ(1P(B)∫B|f|pdP)1/p,‖f‖Lp,λ=sup⁡n≥0sup⁡B∈A(Fn)1P(B)λ(1P(B)∫B|f-Enf|pdP)1/p,
and let(1.2)Lp,λ={f∈Lp0:‖f‖Lp,λ<∞},Lp,λ={f∈Lp0:‖f‖Lp,λ<∞},
where Lp0 is the set of all f∈Lp such that E0f=0.We give basic properties of martingale Morrey-Campanato spaces and compare these spaces with martingale Lipschitz spaces introduced by Weisz [7]. It is well known, in harmonic analysis, that Campanato spaces contain Lipschitz spaces as special cases. Recently, martingale Campanato spaces were introduced in [8] as generalization of martingale Lipschitz spaces. While our definition of martingale Morrey-Campanato spaces is different from the one in [8], we can prove that our martingale Morrey-Campanato spaces contain martingale Lipschitz spaces by Weisz as special cases, under the assumption that every σ-algebra ℱn is generated by countable atoms.The fractional integrals are very useful tools to analyse function spaces in harmonic analysis. Actually, Hardy and Littlewood [9, 10] and Sobolev [11] investigated the fractional integrals to establish the theory of Lebesgue spaces and Lipschitz spaces. Stein and Weiss [12], Taibleson and Weiss [13], and Krantz [14] also investigated the fractional integrals to establish the theory of Hardy spaces. See also [15]. The Lp-Lq boundedness of the fractional integrals is well known as the Hardy-Littlewood-Sobolev theorem derived from [9–11]. This boundedness has been extended to Morrey-Campanato spaces by Peetre [16] and Adams [17], see also [18]. It is known that Morrey-Campanato spaces contain Lp, BMO, and Lipα as special cases, see for example [16, 19].On the other hand, in martingale theory, Watari [20] and Chao and Ombe [21] proved the boundedness of the fractional integrals for Lp (Hp), BMO, and Lipschitz spaces of the dyadic martingale. In this paper, we also establish the boundedness of fractional integrals as martingale transforms on Morrey-Campanato spaces. Our result generalizes and improves the results in [20, 21].For a martingalef=(fn)n≥0 relative to {ℱn}n≥0, denote its martingale difference by dnf=fn-fn-1 (n≥0, with convention d0f=0). For α>0, we define the fractional integral Iαf=((Iαf)n)n≥0 of f by(1.3)(Iαf)n=∑k=0nbk-1αdkf,
where bk is an ℱk-measurable function such that(1.4)bk(ω)=P(B)fora.e.ω∈BwithB∈A(Fk).
Then ((Iαf)n)n≥0 is a martingale for any martingale f=(fn)n≥0, since each bk is bounded, that is, Iα is a martingale transform introduced by Burkholder [22]. This definition of Iα is an extention of the one in [20, 21] which is for dyadic martingales. We can prove the boundedness of fractional integrals Iα as martingale transforms from Lp to Lq, if 1<p<q<∞ and -1/p+α=-1/q. That is, if a martingale f=(fn)n≥0 is Lp-bounded, then ((Iαf)n)n≥0 is Lq-bounded and the following inequality holds:(1.5)sup⁡n≥0⁡‖(Iαf)n‖Lq≤Csup⁡n≥0‖fn‖Lp,
where C is a positive constant independent of f. Further, we prove the boundedness of fractional integrals Iα as martingale transforms on Morrey-Campanato spaces.To prove the boundedness of fractional integrals we use a different method from [20, 21]. More precisely, under the assumption that every σ-algebra ℱn is generated by countable atoms, we first prove the boundedness of the maximal function, and then we use the pointwise estimate by the maximal function and its boundedness, namely, Hedberg’s method in [23]. We also use the method in [24, 25]. By considering sequences of atoms precisely, we can apply these methods to martingale Morrey-Campanato spaces. From this point of view our assumption seems to be natural to define martingale Morrey-Campanato spaces.We state notation, definitions, and remarks in the next section and give basic properties of Morrey-Campanato spaces in Section3. We prove the boundedness of the maximal function and fractional integrals in Sections 4 and 5, respectively.At the end of this section, we make some conventions. Throughout this paper, we always useC to denote a positive constant that is independent of the main parameters involved but whose value may differ from line to line. Constants with subscripts, such as Cp, is dependent on the subscripts. If f≤Cg, we then write f≲g or g≳f and if f≲g≲f, we then write f~g.


## 2. Notation, Definitions, and Remarks

Recall that(Ω,ℱ,P) is a probability space, and {ℱn}n≥0 a nondecreasing sequence of sub-σ-algebras of ℱ such that ℱ=σ(⋃nℱn). For the sake of simplicity, let ℱ-1=ℱ0. As in Section 1, we always suppose that every σ-algebra ℱn is generated by countable atoms, with denoting by A(ℱn) the set of all atoms in ℱn. We define the fractional integral Iα as a martingale transform by (1.3).For a martingalef=(fn)n≥0 relative to {ℱn}n≥0, the maximal function f* of f is defined by(2.1)fn*=sup⁡0≤m≤n|fm|,f*=sup⁡n≥0|fn|.It is known that ifp∈(1,∞), then any Lp-bounded martingale converges in Lp. Moreover, if f∈Lp, p∈[1,∞), then (fn)n≥0 with fn=Enf is an Lp-bounded martingale and converges to f in Lp (see, e.g., [26]). For this reason a function f∈L1 and the corresponding martingale (fn)n≥0 with fn=Enf will be denoted by the same symbol f. Note also that ∥f∥Lp=supn≥0∥Enf∥Lp. In this case(2.2)fn*=sup⁡0≤m≤n|Emf|,f*=sup⁡n≥0|Enf|forf∈L1.Letℳ be the set of all martingales such that f0=0. For p∈[1,∞], let Lp0 be the set of all f∈Lp such that E0f=0. For any f∈Lp0, let fn=Enf. Then (fn)n≥0 is an Lp-bounded martingale in ℳ. For this reason we can regard Lp0 as a subset of ℳ.In Section1 we have introduced Morrey spaces Lp,λ and Campanato spaces ℒp,λ as the following.Definition 2.1.
Letp∈[1,∞) and λ∈(-∞,∞). For f∈L1, let
(2.3)‖f‖Lp,λ=sup⁡n≥0⁡sup⁡B∈A(Fn)⁡1P(B)λ(1P(B)∫B|f|pdP)1/p,‖f‖Lp,λ=sup⁡n≥0⁡sup⁡B∈A(Fn)1P(B)λ(1P(B)∫B|f-Enf|pdP)1/p,
and define
(2.4)Lp,λ={f∈Lp0:‖f‖Lp,λ<∞},Lp,λ={f∈Lp0:‖f‖Lp,λ<∞}.
Then functionals∥f∥Lp,λ and ∥f∥ℒp,λ are norms on Lp,λ and ℒp,λ, respectively. Note that Lp,λ and ℒp,λ are not always trivial set {0} even if λ>0 and λ>1, respectively. This property is different from classical Morrey-Campanato spaces on ℝn.
The martingalef=(fn)n≥0 is said to be Lp,λ-bounded if fn∈Lp,λ (n≥0) and supn≥0∥fn∥Lp,λ<∞. Similarly, the martingale f=(fn)n≥0 is said to be ℒp,λ-bounded if fn∈ℒp,λ (n≥0) and supn≥0∥fn∥ℒp,λ<∞.Proposition 2.2.
Let1≤p<∞. Let f∈L1 and  (fn)n≥0 be its corresponding martingale with  fn=Enf (n≥0).
(i)
Assume thatλ∈(-∞,0]. If f∈Lp,λ, then (fn)n≥0 is Lp,λ-bounded and
(2.5)‖f‖Lp,λ≥sup⁡n≥0‖fn‖Lp,λ.
Conversely, if (fn)n≥0 is Lp,λ-bounded, then f∈Lp,λ and
(2.6)‖f‖Lp,λ≤sup⁡n≥0‖fn‖Lp,λ.(ii)
Assume thatλ∈(-∞,∞). If f∈ℒp,λ, then (fn)n≥0 is ℒp,λ-bounded and
(2.7)‖f‖Lp,λ≥sup⁡n≥0‖fn‖Lp,λ.
Conversely, if (fn)n≥0 is ℒp,λ-bounded, then f∈ℒp,λ and
(2.8)‖f‖Lp,λ≤sup⁡n≥0‖fn‖Lp,λ.Proof.
(i) Letf∈Lp,λ and n≥0. Fix any B∈A(ℱk). If k≤n, then
(2.9)(∫B|Enf|pdP)1/p≤(∫BEn[|f|p]dP)1/p=(∫B|f|pdP)1/p≤P(B)λ+1/p‖f‖Lp,λ.
If n<k, then taking Bn∈A(ℱn) such that B⊂Bn, we have
(2.10)Enf=1P(Bn)∫BnfdPon⁡Bn.
Hence
(2.11)(1P(B)∫B|Enf|pdP)1/p=(1P(Bn)∫Bn|Enf|pdP)1/p≤(1P(Bn)∫BnEn[|f|p]dP)1/p=(1P(Bn)∫Bn|f|pdP)1/p≤P(Bn)λ‖f‖Lp,λ≤P(B)λ‖f‖Lp,λ.
Therefore, fn=Enf∈Lp,λ and ∥fn∥Lp,λ≤∥f∥Lp,λ for all n≥0. This shows that (fn)n≥0 is a Lp,λ-bounded martingale and
(2.12)sup⁡n≥0‖fn‖Lp,λ≤‖f‖Lp,λ.
Conversely, from the inequality
(2.13)∫B|f|pdP≤liminf⁡n→∞∫B|Enf|pdP∀B∈⋃mA(Fm),
it follows that
(2.14)‖f‖Lp,λ≤sup⁡n≥0‖Enf‖Lp,λ.(ii)
Letf∈ℒp,λ and n≥0. Fix any B∈A(ℱk). If k≤n, then(2.15)(∫B|Enf-Ek[Enf]|pdP)1/p=(∫B|Enf-En[Ekf]|pdP)1/p≤(∫BEn[|f-Ekf|p]dP)1/p=(∫B|f-Ekf|pdP)1/p≤P(B)λ+1/p‖f‖Lp,λ.
If n<k, then Ek[Enf]=Enf. Hence
(2.16)(1P(B)∫B|Enf-Ek[Enf]|pdP)1/p=0.
Therefore, fn=Enf∈ℒp,λ and ∥fn∥ℒp,λ≤∥f∥ℒp,λ for all n≥0. This shows that (fn)n≥0 is a ℒp,λ-bounded martingale and
(2.17)sup⁡n≥0‖fn‖Lp,λ≤‖f‖Lp,λ.
Conversely, from the inequality
(2.18)∫B|f-fk|pdP≤liminf⁡n→∞∫B|En[f-fk]|pdP∀B∈A(Fk),k=1,2,…,
it follows that
(2.19)‖f‖Lp,λ≤sup⁡n≥0‖Enf‖Lp,λ.Remark 2.3.
In general, forf∈Lp,λ (resp., ℒp,λ), Enf does not always converge to f in Lp,λ (resp., ℒp,λ). See Remark 3.7.Remark 2.4.
Ifℱ0={∅,Ω},then Lp,λ⊂Lp and ∥f∥Lp≤∥f∥Lp,λ for f∈Lp,λ.Therefore, if (fn)n≥0 is an Lp,λ-bounded martingale, then it is an Lp-bounded martingale. If 1<p<∞,from the known result it follows that there exists f∈Lp0 such that Enf=fn, n≥0, and (fn)n≥0 converges to f in Lp and a.e. Moreover, we can deduce that f∈Lp,λ, since
(2.20)(∫B|fn(ω)|pdP)1/p≤P(B)λ+1/p‖fn‖Lp,λ≤P(B)λ+1/psup⁡n≥0‖fn‖Lp,λ,B∈⋃mA(Fm).
The stochastic basis {ℱn}n≥0 is said to be regular, if there exists a constant R≥2 such that
(2.21)fn≤Rfn-1
holds for all nonnegative martingales (fn)n≥0.Remark 2.5.
In general,Lp,λ⊂ℒp,λ with ∥f∥ℒp,λ≤2∥f∥Lp,λ. Actually, for any B∈A(ℱn),
(2.22)(1P(B)∫B|f-Enf|pdP)1/p≤(1P(B)∫B|f|pdP)1/p+(1P(B)∫B|Enf|pdP)1/p≤2(1P(B)∫B|f|pdP)1/p.
Moreover, if {ℱn}n≥0 is regular and λ<0, then we can prove that Lp,λ=ℒp,λ with equivalent norms (Theorem 3.1 and Remark 3.2).Remark 2.6.
By definition, if0≥λ≥λ′, then we have that L∞0⊂Lp,λ⊂Lp,λ′ with ∥f∥Lp,λ′≤∥f∥Lp,λ≤∥f∥L∞ and L∞0⊂ℒp,λ⊂ℒp,λ′ with ∥f∥ℒp,λ′≤∥f∥ℒp,λ≤2∥f∥L∞. If λ≤-1/p, then Lp0⊂Lp,λ⊂ℒp,λ with ∥f∥ℒp,λ/2≤∥f∥Lp,λ≤∥f∥Lp.Remark 2.7.
By definition and Remark2.5, if ℱ0={∅,Ω}, then Lp,λ⊂ℒp,λ⊂Lp0 with ∥f∥Lp≤∥f∥ℒp,λ≤2∥f∥Lp,λ.Definition 2.8.
LetBMO=ℒ1,0 and Lipα=ℒ1,α if α>0.
Our definitions of BMO andLipα are different from the ones by Weisz [7]. To compare both we give another definition of martingale Morrey and Campanato spaces.Definition 2.9.
Letp∈[1,∞) and λ∈(-∞,∞).For f∈L1, let
(2.23)‖f‖Lp,λ,F=sup⁡n≥0⁡sup⁡B∈Fn1P(B)λ(1P(B)∫B|f|pdP)1/p,‖f‖Lp,λ,F=sup⁡n≥0sup⁡B∈Fn1P(B)λ(1P(B)∫B|f-Enf|pdP)1/p,
and define(2.24)Lp,λ,F={f∈Lp0:‖f‖Lp,λ,F<∞},Lp,λ,F={f∈Lp0:‖f‖Lp,λ,F<∞}.
Note that the spacesLp,λ,ℱ and ℒp,λ,ℱ can be defined without the assumption that every σ-algebra ℱn is generated by countable atoms.Remark 2.10.
By the definitions we have the relationsLp,λ,ℱ⊂Lp,λ with ∥f∥Lp,λ≤∥f∥Lp,λ,ℱ and ℒp,λ,ℱ⊂ℒp,λ with ∥f∥ℒp,λ≤∥f∥ℒp,λ,ℱ. If λ≥0, then we can prove that Lp,λ,ℱ=Lp,λ and ℒp,λ,ℱ=ℒp,λ with the same norms, respectively (see Proposition 3.8). If -1/p<λ<0, then Lp,λ,ℱ⫋Lp,λ and ℒp,λ,ℱ⫋ℒp,λ in general (see Proposition 3.9).Remark 2.11.
It is known that, if{ℱn}n≥0 is regular and λ≥0, then ℒ1,λ,ℱ=ℒp,λ,ℱ with ∥f∥ℒ1,λ,ℱ≤∥f∥ℒp,λ,ℱ≤Cp∥f∥ℒ1,λ,ℱ for each p∈[1,∞) (see, e.g., [8]).
We also define weak Morrey spaces.Definition 2.12.
Forp∈[1,∞) and λ∈(-∞,∞), let
(2.25)‖f‖WLp,λ=sup⁡n≥0sup⁡B∈A(Fn)1P(B)λsup⁡t>0t(P(B∩{|f|>t})P(B))1/p,
for measurable functions f, and define
(2.26)WLp,λ={f∈L10:‖f‖WLp,λ<∞}.


## 3. Basic Properties of Morrey and Campanato Spaces

In this section we give basic properties of Morrey and Campanato spaces. The following theorem gives the relation between Morrey and Campanato spaces.Theorem 3.1.
Let{ℱn}n≥0 be regular, ℱ0={∅,Ω} and (Ω,ℱ,P) be nonatomic. Let p∈[1,∞). (i)
Ifλ≤-1/p, then Lp,λ=ℒp,λ=Lp0 and
(3.1)12‖f‖Lp,λ≤‖f‖Lp,λ=‖f‖Lp≤‖f‖Lp,λ.(ii)
If-1/p<λ<0, then L∞0⫋Lp,λ=ℒp,λ⫋Lp0 and
(3.2)‖f‖Lp≤‖f‖Lp,λ≤‖f‖L∞,12‖f‖Lp,λ≤‖f‖Lp,λ≤C‖f‖Lp,λ.(iii)
Ifλ=0, then L∞0=Lp,0⫋ℒp,0=BMO and
(3.3)‖f‖Lp,0=‖f‖L∞,‖f‖BMO≤‖f‖Lp,0≤Cp‖f‖BMO.(iv)
Ifλ>0, then {0}=Lp,λ⫋ℒp,λ=Lipλ and
(3.4)‖f‖Lipλ≤‖f‖Lp,λ≤Cp‖f‖Lipλ.Remark 3.2.
We can prove (i) without the assumption that{ℱn}n≥0 is regular or that (Ω,ℱ,P) is nonatomic. In (ii), we can prove that Lp,λ=ℒp,λ and (1/2)∥f∥ℒp,λ≤∥f∥Lp,λ≤C∥f∥ℒp,λ without the assumption that ℱ0={∅,Ω} or that (Ω,ℱ,P) is nonatomic. To show L∞0⫋ℒp,λ in (ii) and (iii), we can replace the condition that (Ω,ℱ,P) is nonatomic by a weaker condition as in Proposition 3.6(ii), which follows from the condition that (Ω,ℱ,P) is nonatomic. In (iv), we need the condition that (Ω,ℱ,P) is nonatomic to show Lp,λ={0}.To prove the theorem we first prove a lemma and two propositions.Lemma 3.3.
Let{ℱn}n≥0 be regular. Then every sequence
(3.5)B0⊃B1⊃⋯⊃Bn⊃⋯,Bn∈A(Fn),
has the following property: for each n≥1,
(3.6)Bn=Bn-1or(1+1R)P(Bn)≤P(Bn-1)≤RP(Bn),
where R is the constant in (2.21).Remark 3.4.
SinceBn∈A(ℱn) is an (ℱn,P)-atom, we always interpret Bn-1⊃Bn as the inclusion modulo null sets, that is, Bn-1⊃Bn means P(Bn∖Bn-1)=0. Therefore, Bn=Bn-1 means P(Bn∖Bn-1)=P(Bn-1∖Bn)=0.Remark 3.5.
By the lemma we see that there existsm such that Bm=Bn for all n≥m, if and only if limn→∞P(Bn)>0.Proof of Lemma3.3.
LetB̃n-1={En-1[χBn]≥1/R}. Then B̃n-1∈ℱn-1. By the regularity we have χBn≤REn-1[χBn]. This shows B̃n-1⊃Bn. In this case, B̃n-1⊃Bn-1⊃Bn, since Bn-1∈A(ℱn-1). From the definition of B̃n-1 it follows that χB̃n-1≤REn-1[χBn]. Then we have
(3.7)P(Bn-1)≤P(B̃n-1)=E[χB̃n-1]≤E[REn-1[χBn]]=RE[χBn]=RP(Bn).
Next we show Bn-1=Bn or (1+1/R)P(Bn)≤P(Bn-1). Suppose that
(3.8)P(Bn-1)<(1+1R)P(Bn).
Then
(3.9)P(Bn-1∖Bn)=P(Bn-1)-P(Bn)<P(Bn)R≤P(Bn-1)R.
Therefore,
(3.10)En-1[χBn-1∖Bn]=P(Bn-1∖Bn)P(Bn-1)χBn-1<1RχBn-1.
From the regularity and the inequality above it follows that
(3.11)χBn-1∖Bn≤REn-1[χBn-1∖Bn]<χBn-1.
This means that Bn-1=Bn.Proposition 3.6.
Let{ℱn}n≥0 be regular,  1≤p<∞ and  λ>-1/p.
(i)
For a sequenceB0⊃B1⊃⋯⊃Bk⊃⋯, Bk∈A(ℱk), let f0=0 and
(3.12)fn=∑k=1nP(Bk)λ(P(Bk-1)P(Bk)χBk-χBk-1),n≥1.Then f=(fn)n≥0 is a martingale in ℳ and in ℒp,λ.(ii)
Let0≥λ>λ′>-1/p. If there exists a sequence B0⊃B1⊃⋯⊃Bk⊃⋯, Bk∈A(ℱk) and limk→∞P(Bk)=0, then L∞0⫋ℒp,λ⫋ℒp,λ′. If ℱ0={∅,Ω} also, then L∞0⫋ℒp,λ⫋ℒp,λ′⫋Lp0.Proof.
(i) By the definition of the sequence(fn)n≥0, we have
(3.13)En-1[dnf]=P(Bn)λ(P(Bn-1)P(Bn)En-1[χBn]-χBn-1)=0,
for every n≥1. Hence, we obtain that (fn)n≥0 is a martingale.
We next show that the sequence(fn)n≥0 converges in Lp. If limk→∞P(Bk)>0 then the convergence is clear by Remark 3.5. We assume that limk→∞P(Bk)=0. Then, by Lemma 3.3, we can take a sequence of integers 0=k0<k1<⋯<kj<⋯ that satisfies
(3.14)(1+1R)P(Bkj)≤P(Bkj-1)≤RP(Bkj),
and Bkj-1=Bk if kj-1≤k<kj. In this case we can write
(3.15)fn=∑kj≤nP(Bkj)λ(P(Bkj-1)P(Bkj)χBkj-χBkj-1).
Using (3.14) and the assumption λ>-1/p, we have
(3.16)∑kj>nP(Bkj)λ‖P(Bkj-1)P(Bkj)χBkj-χBkj-1‖Lp≤∑kj>nP(Bkj)λ(R‖χBkj‖Lp+‖χBkj-1‖Lp)≤2R∑kj>nP(Bkj)λ+1/p≤2R∑j=0∞(1+1R)-(λ+1/p)jP(Bn)λ+1/p∼P(Bn)λ+1/p.
Therefore, (fn)n≥0 converges in Lp. We denote by f the limit of (fn)n≥0.
We can also deduce from (3.16) that
(3.17)(1P(Bn)∫Bn|f-Enf|pdP)1/p≲P(Bn)λ.
On the other hand, for B∈A(ℱn), we have
(3.18)(f-Enf)χB={f-Enf,(B=Bn),0,(B≠Bn).
Combining (3.17) and (3.18), we have ∥f∥ℒλ,p≲1, that is, we get f∈ℒp,λ.
(ii) First we showL∞0⫋ℒp,λ⫋ℒp,λ′. By Remark 2.6 we have L∞0⊂ℒp,λ⊂ℒp,λ′. Then we need to show ℒp,0∖L∞0≠∅ and ℒp,λ′∖ℒp,λ≠∅.
We considerfn in (3.12) for the sequence  Bk, k=0,1,…. Then we can write
(3.19)fn=∑kj≤n(P(Bkj-1)P(Bkj)χBkj-χBkj-1)
for λ=0 and we have f=(fn)n≥0∈ℒp,0. On the other hand, for a.e. ω∈Bkj∖Bkj+1,
(3.20)f=fkj+1=fkj-χBkj=∑i=0j(P(Bki-1)P(Bki)-1)-1≥1R(j+1)-1.
Thenf∉L∞0 and ℒp,0∖L∞0≠∅.
Next, let(3.21)gn=∑kj≤nP(Bkj)λ′(P(Bkj-1)P(Bkj)χBkj-χBkj-1).
Then g=(gn)n≥0∈ℒp,λ′. On the other hand, since
(3.22)|g-Ekj-1g|=P(Bkj)λ′on⁡Bkj-1∖Bkj,
we have
(3.23)1P(Bkj-1)λ(1P(Bkj-1)∫Bkj-1|g-Ekj-1g|pdP)1/p≥1P(Bkj-1)λ(1P(Bkj-1)∫Bkj-1∖Bkj|g-Ekj-1g|pdP)1/p≥P(Bkj)λ′P(Bkj-1)λ(P(Bkj-1∖Bkj)P(Bkj-1))1/p≥Rλ′(1+R)-λ′-1/pP(Bkj-1)λ′-λ⟶∞asj⟶∞,
since λ′<λ. This shows ℒp,λ′∖ℒp,λ≠∅.
Finally, ifℱ0={∅,Ω} also, then by Remark 2.7 and ℒp,λ′∖ℒp,λ≠∅ we have ℒp,λ⫋Lp0. This shows the conclusion.Remark 3.7.
In Proposition3.6, f=(fn)n≥0 in (3.12) converges in Lp as in the above proof. Moreover, the limit belongs to both ℒp,λ and Lp,λ when -1/p<λ<0, since we will show that Lp,λ=ℒp,λ in the proof of Theorem 3.1. However, it converges in neither ℒp,λ nor Lp,λ. Actually, by a similar calculation to g=(gn)n≥0 in the above proof, we have
(3.24)|(f-fn)-Ekj-1(f-fn)|=|f-Ekj-1f|=P(Bkj)λon⁡Bkj-1∖Bkj,
for n≤kj-1, and then
(3.25)1P(Bkj-1)λ(1P(Bkj-1)∫Bkj-1|f-Ekj-1f|pdP)1/p≥Rλ(1+R)-λ-1/p.
By Remark 2.5, we have
(3.26)2‖f-fn‖Lp,λ≥‖f-fn‖Lp,λ≥Rλ(1+R)-λ-1/p.
This shows that (fn)n≥0 converges in neither ℒp,λ nor Lp,λ.Proposition 3.8.
Let1≤p<∞.(i)
Ifℱ0={∅,Ω} and λ≤-1/p, then Lp0=Lp,λ=Lp,λ,ℱ=ℒp,λ=ℒp,λ,ℱ with(3.27)‖f‖Lp,λ,F=‖f‖Lp,λ=‖f‖Lp≤‖f‖Lp,λ≤‖f‖Lp,λ,F≤2‖f‖Lp,λ,F.(ii)
Ifλ≥0, then Lp,λ=Lp,λ,ℱ⊂ℒp,λ=ℒp,λ,ℱ⊂Lp0 with(3.28)‖f‖Lp≤‖f‖Lp,λ,F=‖f‖Lp,λ≤‖f‖Lp,λ,F=‖f‖Lp,λ.Proof.
(i) Letλ≤-1/p. For any f∈Lp0 and any B∈ℱn,
(3.29)1P(B)λ(1P(B)∫B|f|pdP)1/p=P(B)-λ-1/p(∫B|f|pdP)1/p≤P(Ω)-λ-1/p(∫Ω|f|pdP)1/p=‖f‖Lp=P(Ω)-λ-1/p(∫Ω|f-E0f|pdP)1/p.
Then by the definition of the norms and the assumption ℱ0={∅,Ω} we see that ∥f∥Lp,λ,ℱ=∥f∥Lp,λ=∥f∥Lp≤∥f∥ℒp,λ≤∥f∥ℒp,λ,ℱ. By the same observation as Remark 2.5 we have ∥f∥ℒp,λ,ℱ≤2∥f∥Lp,λ,ℱ.
(ii) Letλ≥0. By Remark 2.10 we need to show only Lp,λ⊂Lp,λ,ℱ with ∥f∥Lp,λ,ℱ≤∥f∥Lp,λ and ℒp,λ⊂ℒp,λ,ℱ⊂Lp0 with ∥f∥Lp≤∥f∥ℒp,λ,ℱ≤∥f∥ℒp,λ.
Letf∈Lp,λ. For any B∈ℱn, there exists a sequence of atoms Bℓ∈A(ℱn), ℓ=1,2,…, such that B=∪ℓBℓ and P(B)=∑ℓP(Bℓ). Then
(3.30)∫B|f|pdP=∑l∫Bl|f|pdP≤∑lP(Bl)λp+1‖f‖Lp,λp≤P(Bl)λp+1‖f‖Lp,λp,
since λp+1≥1. Therefore f∈Lp,λ,ℱ and ∥f∥Lp,λ,ℱ≤∥f∥Lp,λ. Similarly, we have ∥f∥ℒp,λ,ℱ≤∥f∥ℒp,λ for f∈ℒp,λ. By the definition of ℒp,λ,ℱ norm we have ∥f∥Lp=∥f-E0f∥Lp≤P(Ω)λ+1/p∥f∥ℒp,λ,ℱ=∥f∥ℒp,λ,ℱ for f∈ℒp,λ,ℱ.Proof of Theorem3.1.
(i) We have the conclusion by Proposition3.8 without the assumption that {ℱn}n≥0 is regular or that (Ω,ℱ,P) is nonatomic.
(ii) By Proposition3.6 we only need to prove Lp,λ=ℒp,λ with (1/2)∥f∥ℒp,λ≤∥f∥Lp,λ≤C∥f∥ℒp,λ. The first norm inequality follows from Remark 2.5. We show the second one. Note that we do not need the assumption that ℱ0={∅,Ω} or that (Ω,ℱ,P) is nonatomic.
Letf∈ℒp,λ. Then, for any B∈A(ℱn),
(3.31)(1P(B)∫B|f|pdP)1/p≤(1P(B)∫B|f-Enf|pdP)1/p+|1P(B)∫Bf(ω)dP|≤P(B)λ‖f‖Lp,λ+|1P(B)∫Bf(ω)dP|,
since
(3.32)Enf=1P(B)∫Bf(ω)dPon⁡B.
If B∈ℱ0, then Enf=0 on B. Assume that B∉ℱ0. By Lemma 3.3 we can choose Bkj∈A(ℱkj), 0=k0<k1<⋯<km≤n, such that Bk0⊃Bk1⊃Bk2⊃⋯⊃Bkm=B and that (1+1/R)P(Bkj)≤P(Bkj-1)≤RP(Bkj). Then, since
(3.33)1P(Bk0)∫Bk0f(ω)dP=0,
we have
(3.34)1P(B)∫Bf(ω)dP=1P(B)∫Bf(ω)dP-1P(Bk0)∫Bk0f(ω)dP=∑j=1m(1P(Bkj)∫Bkjf(ω)dP-1P(Bkj-1)∫Bkj-1f(ω)dP)=∑j=1m1P(Bkj)∫Bkj[f-Ekj-1f](ω)dP.
By Hölder’s inequality and the assumption λ<0 we have
(3.35)|1P(B)∫Bf(ω)dP|≤∑j=1m(1P(Bkj)∫Bkj|f-Ekj-1f|pdP)1/p≲∑j=1m(1P(Bkj-1)∫Bkj-1|f-Ekj-1f|pdP)1/p≤∑j=1mP(Bkj-1)λ‖f‖Lp,λ≤∑j=1m((1+1R)m-j+1P(Bkm))λ‖f‖Lp,λ~P(B)λ‖f‖Lp,λ.
Therefore we have ∥f∥Lp,λ≲∥f∥ℒp,λ.
(iii) Letλ=0. By Remark 2.6 and Proposition 3.6 we have L∞0⊂Lp,0 with ∥f∥Lp,0≤∥f∥L∞ and L∞0⫋ℒp,0.
Next we showLp,0⊂L∞0 and ∥f∥L∞≤∥f∥Lp,0.
Letf∈Lp,0 and f≠0 a.e. Take a positive number r such that P(|f|>r)>0. For any ϵ>0, there exists n and B∈ℱn such that
(3.36)P(B∩{|f|>r})>(1-ϵ)P(B),
because ℱ is generated by ∪nℱn. For the above B, we can take a sequence of atoms Bℓ∈A(ℱn), ℓ=1,2,…, such that B=∪ℓBℓ and P(B)=∑ℓP(Bℓ). Hence, by the pigeonhole principle, there exists B′∈A(ℱn) such that
(3.37)P(B′∩{|f|>r})>(1-ϵ)P(B′).
Therefore, we have
(3.38)‖f‖Lp,0p≥1P(B′)∫B′|f|pdP≥1P(B′)∫B′∩{|f|>r}|f|pdP≥P(B′∩{|f|>r})P(B′)rp≥(1-ϵ)rp.
This shows that P(|f|>r)>0 implies ∥f∥Lp,0≥r. Then we have the conclusion.
By Proposition3.8 and Remark 2.11 we have that BMO(=ℒ1,λ)=ℒp,λ with equivalent norms.
(iv) Letλ>0. For f∈Lp,λ, we suppose that there exists r>0 such that P(|f|>r)>0. Then, for any ϵ>0, we have the same estimate as (3.37). Moreover, we can decompose B′ in (3.37) to atoms in A(ℱn+1) and we have the same estimate as (3.37) for some atom in A(ℱn+1). Therefore, we can take an atom B̃′ in A(ℱn+m) for large enough m such that B̃′ satisfies (3.37) and P(B̃′)<ϵ. Hence we have
(3.39)‖f‖Lp,λp≥1P(B′̃)1+λp∫B̃′|f|pdP≥(1-ϵ)rpP(B′̃)λp≥(1-ϵ)rpελp,
that is, r≤ϵλ/(1-ϵ)1/p∥f∥Lp,λ. This contradicts that r>0. Then f=0 a.e. and Lp,λ={0}. By Proposition 3.6 we have that {0}⫋ℒp,λ.
Finally, by Proposition3.8 and Remark 2.11 we have that ℒ1,λ=ℒp,λ with equivalent norms.Next we prove thatLp,λ,ℱ⫋Lp,λ and ℒp,λ,ℱ⫋ℒp,λ in general by an example.Proposition 3.9.
Let(Ω,ℱ,P) be as follows:
(3.40)Ω=[0,1),A(Fn)={In,j=[j2-n,(j+1)2-n):j=0,1,…,2n-1},(3.41)Fn=σ(A(Fn)),F=σ(⋃nFn),P=theLebesguemeasure.
If -1/p<λ<0, then Lp,λ,ℱ⫋Lp,λ and ℒp,λ,ℱ⫋ℒp,λ.Proof.
We constructf such that
(3.42)f∈Lp,λ∖Lp,λ,F,f∈Lp,λ∖Lp,λ,F.
Step 1.
Denote the characteristic function ofIn,j by χn,j and let
(3.43)fn=∑j=02n-1fn+m,2mj,fn,j=P(In,j)λ(χn+1,2j-χn+1,2j+1),
where we choose m such that
(3.44)P(In+m,0)pλ+1≤P(In,0).
Note thatIn,j=In+1,2j∪In+1,2j+1 and |fn,j|=P(In,0)λχn,j.
Ifk≤n+m, then Ekfn=0, and |fn-Ekfn|=|fn|=P(In+m,0)λ∑j=12n-1χn+m,2mj. Hence(3.45)(1P(Ik,l)∫Ik,l|fn-Ekfn|pdP)1/p=(1P(Ik,l)∫Ik,l|fn|pdP)1/p=P(In+m,0)λ(∑j:In+m,2mj⊂Ik,lP(In+m,2mj)P(Ik,l))1/p.
Ifk≤n, then the number of the elements of {j:In+m,2mj⊂Ik,ℓ} is the same as of {j:In,j⊂Ik,ℓ}. Hence
(3.46)(1P(Ik,l)∫Ik,l|fn-Ekfn|pdP)1/p=(1P(Ik,l)∫Ik,l|fn|pdP)1/p≤P(In+m,0)λ(P(In+m,0)P(In,0))1/p≤1≤P(Ik,l)λ,
where we use (3.44) and λ<0 for the last two inequalities. If n<k≤n+m, then the number of the elements of {j:In+m,2mj⊂Ik,ℓ} is one at most. Hence
(3.47)(1P(Ik,l)∫Ik,l|fn-Ekfn|pdP)1/p=(1P(Ik,l)∫Ik,l|fn|pdP)1/p≤P(In+m,0)λ(P(In+m,0)P(Ik,l))1/p≤P(Ik,l)λ,
where we use -1/q<λ<0 for the last inequality. In the above, if Ik,ℓ=In+m,0, then the equality holds.
Ifk>n+m, then fn-Ekfn=0 and
(3.48)(1P(Ik,l)∫Ik,l|fn|pdP)1/p≤P(In+m,0)λ≤P(Ik,0)λ.
Therefore,
(3.49)fn∈Lp,λ∩Lp,λ,‖fn‖Lp,λ=‖fn‖Lp,λ=1.
On the other hand, for the set B=⋃j=02n-1In+m,2mj∈ℱn+m,
(3.50)(1P(B)∫B|fn-En+mfn|pdP)1/p=(1P(B)∫B|fn|pdP)1/p=P(In+m,0)λ=(2-nP(B))λ.
Therefore,∥fn∥Lp,λ,ℱ,∥fn∥ℒp,λ,ℱ≥2-nλ→∞ as n→∞.Step 2.
Letfn be as in Step 1. If k<n, then, by the same observation as Step 1 we also have that
(3.51)fnχk,l∈Lp,λ∩Lp,λ,‖fnχk,l‖Lp,λ=‖fnχk,l‖Lp,λ=1.
Moreover, if k<n, then, for the set B=(⋃j=02n-1In+m,2mj)∩Ik,ℓ∈ℱn+m,
(3.52)(1P(B)∫B|fnχk,l-En+m[fnχk,l]|pdP)1/p=(1P(B)∫B|fnχk,l|pdP)1/p=P(In+m,0)λ=(2-n+kP(B))λ.
Therefore, ∥fnχk,ℓ∥Lp,λ,ℱ,∥fnχk,ℓ∥ℒp,λ,ℱ≥2(-n+k)λ.Step 3.
Letfn be as in Step 1 and let
(3.53)f=∑n=1∞2nλ/2f2nχn,1.
Note that ∑n=1∞χn,1≡1. Then
(3.54)‖f‖Lp,λ,‖f‖Lp,λ≤∑n=1∞2nλ/2=2λ/21-2λ/2.
On the other hand, by the same observation as Step 2, we have that
(3.55)‖f‖Lp,λ,F,‖f‖Lp,λ,F≥2nλ/2×2(-2n+n)λ=2-nλ/2,
for all n. This shows (3.42).At the end of this section we prove the relation of∥f∥L1,λ and ∥f∥WLp. Recall that(3.56)‖f‖WLp=sup⁡t>0tP(|f|>t)1/p.The following is well known for classical Morrey spaces onℝn. We give a proof for convenience, though it is the same as the proof for classical Morrey spaces on ℝn.Proposition 3.10.
If1<q<p<∞ and  -1/p=λ, then
(3.57)‖f‖L1,λ≤‖f‖Lq,λ≤C‖f‖WLp.Proof.
The first inequality followed from Hölder’s inequality. To show the second inequality, we assume that∥f∥WLp=1 and prove that ∥f∥Lq,λ≤C. From ∥f∥WLp=1, we have P(|f|>t)≤t-p. For any atom B∈A(ℱn), let η=P(B)λ=P(B)-1/p and
(3.58)f=fη+fη,fη(ω)={f(ω),|f(ω)|>η,0,|f(ω)|≤η.
Then
(3.59)1P(B)qλ+1∫B|fη(ω)|qdP≤1P(B)qλ+1∫0∞P(|fη|>t)(qtq-1)dt≤1P(B)qλ+1(∫0ηP(|fη|>η)(qtq-1)dt+∫η∞t-p(qtq-1)dt)≤1P(B)qλ+1(1+qp-q)ηq-p=pp-q.
Since |fη(ω)|≤η we have
(3.60)1P(B)λ(1P(B)∫B|fη(ω)|qdP)1/q≤ηP(B)λ=1.
We get the conclusion.


## 4. Maximal Function

It is known as Doob’s inequality that (see for example [5, Pages 20-21])(4.1)‖f*‖Lp≤pp-1‖f‖Lp,f∈Lp(p>1),(4.2)‖f*‖WL1≤‖f‖L1,f∈L1.In this section we extend (4.1) and (4.2) to Morrey norms. Note that we do not need the regularity of the stochastic basis {ℱn}n≥0.Theorem 4.1.
Let1≤p<∞ and  -1/p≤λ<0. Then, for  f∈L1,
(4.3)‖f*‖Lp,λ≤Cp‖f‖Lp,λ,ifp>1,‖f*‖WL1,λ≤C1‖f‖L1,λ,ifp=1.Proof.
Case  1 (p>1). For any B∈A(ℱm) and m≥0, let f=g+h and g=fχB. Then, using (4.1), we have
(4.4)∫B(g*)pdP≤∫Ω(g*)pdP≲∫Ω|g|pdP=∫B|f|pdP.
Hence
(4.5)1P(B)λ(1P(B)∫B(g*)pdP)1/p≲‖f‖Lp,λ.
Next, takeBn∈A(ℱn), n=0,1,…,m, such that B=Bm⊂Bm-1⊂…⊂B0. Then, for a.e. ω∈B,
(4.6)Enh(ω)={0,(n≥m),1P(Bn)∫BnhdP,(n<m).
If n<m, then
(4.7)|Enh(ω)|≤(1P(Bn)∫Bn|h|pdP)1/p≤P(Bn)λ‖f‖Lp,λ≤P(B)λ‖f‖Lp,λ,
since λ<0. Hence
(4.8)h*≤P(B)λ‖f‖Lp,λon⁡B.
Then(4.9)1P(B)λ(1P(B)∫B(h*)pdP)1/p≤‖f‖Lp,λ.
By (4.5), (4.9), and the inequality f*≤g*+h*, we have
(4.10)1P(B)λ(1P(B)∫B(f*)pdP)1/p≲‖f‖Lp,λ,
which shows the conclusion.
Case  2 (p=1). Let g and h be as in Case 1. Using (4.2), we have, for all t>0,
(4.11)tP(B∩{g*>t})≤tP(g*>t)≤∫Ω|g|dP=∫B|f|dP
and then
(4.12)1P(B)λtP(B∩{g*>t})P(B)≤‖f‖L1,λ.
We also have (4.8) for the case p=1. Then
(4.13)1P(B)λtP(B∩{h*>t})P(B)≤‖f‖L1,λ.
Therefore we have the conclusion.


## 5. Fractional Integrals

In this section we establish the boundedness of the fractional integrals. To do this we first prove norm inequalities for functions, and then we get the boundedness ofIα as a martingale transform.For normed spacesM1 and M2 of functions, we denote by B(M1,M2) the set of all bounded martingale transforms from M1 to M2, that is, Iα∈B(M1,M2) means that(5.1)sup⁡n≥0‖(Iαf)n‖M2≤Csup⁡n≥0‖fn‖M1,
for all M1-bounded martingales f=(fn)n≥0.We state our results in Section5.1. To prove Theorems 5.1 and 5.5 we show the pointwise estimate for Iαf with the assumptions ∥f∥Lp<∞ and ∥f∥Lp,λ<∞, respectively. To avoid repetition we prove first Theorem 5.5 in Section 5.2 and then Theorem 5.1 in Section 5.3. The proof of Theorem 5.8 is in Section 5.4.


### 5.1. Boundedness of Fractional Integrals

The following is forLp-Lq boundedness.Theorem 5.1.
Assume that{ℱn}n≥0 is regular. Let  1≤p<q<∞,  -1/p+α=-1/q. Then, for  f∈L1,
(5.2)‖(Iαf)*‖Lq≤C‖f‖Lp,ifp>1,(5.3)‖(Iαf)*‖WLq≤C‖f‖L1,ifp=1.Remark 5.2.
Letf=(fn)n≥0 be an L1-bounded martingale. For each L1-function fm, m≥0, consider the corresponding martingale fm=(En[fm])n≥0=(fmin(n,m))n≥0. Then (Iαfm)n=(Iαf)n  for m≥n. Therefore, from (5.3) it follows that, for m≥n,
(5.4)‖(Iαf)n‖WLq=‖(Iαfm)n‖WLq≲‖fm‖L1≤sup⁡n≥0‖fn‖L1.
This shows that
(5.5)sup⁡n≥0‖(Iαf)n‖WLq≤Csup⁡n≥0‖fn‖L1.Remark 5.3.
Let a martingalef=(fn)n≥0 be L1-bounded. Since WLq⊂Lq1 and ∥(Iαf)n∥Lq1≤∥(Iαf)n∥WLq for 1<q1<q, from Remark 5.2 it follows that the martingale ((Iαf)n)n≥0isLq1-bounded and that it converges in Lq1. Denote this limit by Iαf.Then, Iαf=∑k=0∞bk-1αdkf and En[Iαf]=(Iαf)n.
Ifp>1, then the same observation as in Remark 5.2 with (5.2) shows that
(5.6)sup⁡n≥0‖(Iαf)n‖Lq≤Csup⁡n≥0‖fn‖Lp.
Hence we have the following corollary.Corollary 5.4.
Assume that{ℱn}n≥0 is regular. Let  1≤p<q<∞,  -1/p+α=-1/q. Then
(5.7)Iα∈B(Lp,Lq),ifp>1,Iα∈B(L1,WLq),ifp=1.For Morrey norms, one has the following.Theorem 5.5.
Assume that{ℱn}n≥0 is regular. Let  1≤p<∞,  -1/p≤λ<0, and  μ=λ+α<0. Then, for  f∈L1,
(5.8)‖(Iαf)*‖Lq,μ≤C‖f‖Lp,λ,ifp>1,1≤q≤(λ/μ)p,(5.9)‖(Iαf)*‖Lq,μ≤C‖f‖L1,λ,ifp=1,1≤q<(λ/μ)p,(5.10)‖(Iαf)*‖WLq,μ≤C‖f‖L1,λ,ifp=1,q=(λ/μ)p.Note that Theorem5.1 is not a corollary of Theorem 5.5, since ∥f∥Lp≲∥f∥Lp,-1/p≤∥f∥Lp in general.Remark 5.6.
In order to prove (5.8) it suffices to prove it in the case where q=(λ/μ)p, since ∥f∥Lq1,λ≤∥f∥Lq,λ for q1≤q by Hölder’s inequality. The inequality (5.9) follows from (5.10), since ∥f∥Lq1,λ≲∥f∥WLq,λ for q1<q.By the same observation as in Remark5.2 we have the following.Corollary 5.7.
Assume that{ℱn}n≥0 is regular. Let  1≤p<∞,  -1/p≤λ<0, and  μ=λ+α<0. Then
(5.11)Iα∈B(Lp,λ,Lq,μ),ifp>1,1≤q≤(λ/μ)p,Iα∈B(L1,λ,Lq,μ),ifp=1,1≤q<(λ/μ)p,Iα∈B(L1,λ,WLq,μ),ifp=1,q=(λ/μ)p.For Campanato spaces, one has the following.Theorem 5.8.
Assume that{ℱn}n≥0 is regular. Let  1≤p<∞,  λ≥-1/p,  1≤q<∞, and  μ=λ+α≥0. Then
(5.12)‖Iαf‖Lq,μ≤C‖f‖Lp,λ,f∈L1,
and
(5.13)Iα∈B(Lp,λ,Lq,μ),
where Iαf in (5.12) denotes the limit function described in Remark 5.3, for f=(Enf)n≥0, f∈L1.Remark 5.9.
Ifμ<0, then, by Theorem 3.1 (ii), the inequalities (5.8) and (5.9) in Theorem 5.5 hold with Lp,λ and Lq,μ replaced by ℒp,λ and ℒq,μ, respectively.
If1<p<∞ and -1/p=λ, then ∥f∥ℒ1,λ~∥f∥L1,λ≲∥f∥WLp≲∥f∥Lp by Remark 2.5 and Proposition 3.10. Since Lp⫋L1,λ in general, the following is an improvement of the results in [21].Corollary 5.10.
Assume that{ℱn}n≥0 is regular. Then, for  f∈L1,
(5.14)‖Iαf‖BMO≤C‖f‖L1,-1/p,if-1p+α=0,1<p<∞,‖Iαf‖Lipβ≤C‖f‖L1,-1/p,if-1p+α=β>0,1<p<∞,‖Iαf‖Lipα≤C‖f‖BMO,‖Iαf‖Lipγ≤C‖f‖Lipβ,ifβ+α=γ,β>0,(5.15)Iα∈B(L1,-1/p,BMO),if-1p+α=0,1<p<∞,Iα∈B(L1,-1/p,Lipβ),if-1p+α=β>0,1<p<∞,Iα∈B(BMO,Lipα),Iα∈B(Lipβ,Lipγ),ifβ+α=γ,β>0.


### 5.2. Proof of Theorem5.5

By Remark5.6 we only need to prove (5.8) and (5.10) in the case q=(λ/μ)p. So we always assume that q=(λ/μ)p in this proof.First we show the pointwise estimate(5.16)(Iαf)*(ω)≤Cf*(ω)μ/λ‖f‖Lp,λ-α/λ,whereC is a positive constant independent of ω. To do this we prove that, for any m≥1 and any Bm∈A(ℱm),(5.17)|(Iαf)m(ω)|≤Cf*(ω)μ/λ‖f‖Lp,λ-α/λ,ω∈Bm.TakeBk∈A(ℱk), 0≤k<m, such that Bm⊂Bm-1⊂…⊂B0 and let(5.18)K={k:0<k≤m,Bk≠Bk-1}={k1,k2,…,kl},where0=k0<k1<k2<…<kℓ. Then, by Lemma 3.3,(5.19)(1+1R)bkj≤bkj-1≤Rbkjon⁡Bm.Hence, we can write(5.20)(Iαf)m=∑0<kj≤mbkj-1αdkjf=∑j=1lbkj-1αdkjfon⁡Bm,sincebk=bk-1 and dkf=0 for k∉K. Note that for ω∈Bm,(5.21)|dkjf(ω)|=|fkj(ω)-fkj-1(ω)|≤|fkj(ω)|+|fkj-1(ω)|=|1P(Bkj)∫BkjfdP|+|1P(Bkj-1)∫Bkj-1fdP|≤(1P(Bkj)∫Bkj|f|pdP)1/p+(1P(Bkj-1)∫Bkj-1|f|pdP)1/p≤(P(Bkj)λ+P(Bkj-1)λ)‖f‖Lp,λ≤(1+(1R)λ)bkj-1(ω)λ‖f‖Lp,λ.Then, forω∈Bm and for 0<n≤m,(5.22)|∑k=0nbk-1(ω)αdkf(ω)|≤(1+(1R)λ)∑0<kj≤nbkj-1(ω)μ‖f‖Lp,λ≤1+(1/R)λ1-(1+1/R)μbn(ω)μ‖f‖Lp,λ.On the other hand, for0≤n<m, letting j(n)=min{j:n<kj}, we have, for ω∈Bm,(5.23)∑k=n+1mbk-1(ω)αdkf(ω)=∑j=j(n)lbkj-1(ω)αdkjf(ω)=∑j=j(n)lbkj-1(ω)αfkj(ω)-∑j=j(n)lbkj-1(ω)αfkj-1(ω)=bkl-1(ω)αfkl(ω)+∑j=j(n)l-1(bkj-1(ω)α-bkj(ω)α)fkj(ω)-bkj(n)-1(ω)αfkj(n)-1(ω),(5.24)|∑k=n+1mbk-1(ω)αdkf(ω)|≤bkl-1(ω)αf*(ω)+∑j=j(n)l-1|bkj-1(ω)α-bkj(ω)α|f*(ω)+bkj(n)-1(ω)αf*(ω)≤2bkj(n)-1(ω)αf*(ω)=2bn(ω)αf*(ω).Here, let(5.25)Ω1={ω∈Ω:(f*(ω)‖f‖Lp,λ)1/λ≤b0(ω)},Ω2=Ω∖Ω1.Ifω∈Ω1∩Bm and(5.26)(f*(ω)‖f‖Lp,λ)1/λ≤bm(ω),then, by (5.22) we have(5.27)|(Iαf)m(ω)|≲bm(ω)μ‖f‖Lp,λ≤(f*(ω)‖f‖Lp,λ)μ/λ‖f‖Lp,λ=f*(ω)μ/λ‖f‖Lp,λ-α/λ,sinceμ<0 and 1-μ/λ=-α/λ. If ω∈Ω1∩Bm and(5.28)bm(ω)<(f*(ω)‖f‖Lp,λ)1/λ,choosingn such that(5.29)R-1bn(ω)<(f*(ω)‖f‖Lp,λ)1/λ≤bn(ω),we have by (5.22) and (5.24)(5.30)|(Iαf)m(ω)|≲bn(ω)μ‖f‖Lp,λ+bn(ω)αf*(ω)≲(f*(ω)‖f‖Lp,λ)μ/λ∥f∥Lp,λ+(f*(ω)‖f‖Lp,λ)α/λf*(ω)≲f*(ω)μ/λ‖f‖Lp,λ-α/λ.Ifω∈Ω2∩Bm, then by (5.24) we have(5.31)|(Iαf)m(ω)|=|∑k=1mbk-1(ω)αdkf(ω)|≲b0(ω)αf*(ω)≤(f*(ω)‖f‖Lp,λ)α/λf*(ω)=f*(ω)μ/λ‖f‖Lp,λ-α/λ.Therefore, we have (5.17).Next, applying the boundedness of the maximal function (Theorem4.1), we show(5.32)(1P(B)∫B[(Iαf(ω))*]qdP)1/q≲P(B)μ‖f‖Lp,λ,ifp>1,(5.33)sup⁡t>0t(P(B∩{(Iαf)*>t})P(B))1/q≲P(B)μ‖f‖L1,λ,ifp=1,for anyn and any B∈A(ℱn). To show (5.32), using (5.16) and the relations qμ/λ=p and -α/λ=1-p/q, we have(5.34)(1P(B)∫B[(Iαf(ω))*]qdP)1/q≲(1P(B)∫Bf*(ω)qμ/λdP)1/q‖f‖Lp,λ-α/λ=(1P(B)∫Bf*(ω)pdP)(1/p)(p/q)‖f‖Lp,λ1-p/q.Applying the boundedness of the maximal function, we have(5.35)(1P(B)∫Bf*(ω)pdP)(1/p)(p/q)≤(P(B)λ‖f*‖Lp,λ)p/q≲P(B)μ‖f‖Lp,λp/q.Then we have (5.32). To show (5.33), using (5.16) and the relations p=1, μ/λ=1/q and -α/λ=1-1/q, we have(5.36)sup⁡t>0t(P(B∩{(Iαf)*>t})P(B))1/q≲sup⁡t>0t(P(B∩{(f*)μ/λ‖f‖L1,λ-α/λ>t})P(B))1/q=sup⁡t>0t(P(B∩{(f*)1/q∥f∥L1,λ1-1/q>t})P(B))1/q=sup⁡t>0t(P(B∩{(f*)1/q>t})P(B))1/q‖f‖L1,λ1-1/q=(sup⁡t>0tP(B∩{(f*)>t})P(B))1/q‖f‖L1,λ1-1/q.Applying the boundedness of the maximal function, we have(5.37)(sup⁡t>0tP(B∩{(f*)>t})P(B))1/q≤(P(B)λ‖f*‖WL1,λ)1/q≲P(B)μ‖f‖Lp,λ1/q.Then we have (5.33). The proof is complete.


### 5.3. Proof of Theorem5.1

Letλ=-1/p and μ=-1/q. Then μ/λ=p/q and -α/λ=(1/p-1/q)p=1-p/q. In this case the pointwise estimate (5.16) implies(5.38)(Iαf)*(ω)≤Cf*(ω)p/q‖f‖Lp,-1/p1-p/q≤Cf*(ω)p/q‖f‖Lp1-p/q.Applying the boundedness of the maximal function onLp, we get the conclusion.


### 5.4. Proof of Theorem5.8

Note that, forf∈L1, ∥f∥ℒp,λ=∥f-E0f∥ℒp,λ and Iαf=Iα(f-E0f). Then it is sufficient to prove (5.12) only for f∈L10. Therefore, (5.12) follows from (5.13) and Proposition 2.2(ii).By Hölder’s inequality we have∥f∥ℒ1,λ≤∥f∥ℒp,λ for p≥1. Furthermore, by Theorem 3.1(iii) and (iv) we have ∥Iαf∥ℒq,μ~∥Iαf∥ℒ1,μ for q≥1. So we only need to prove that(5.39)sup⁡m≥0‖(Iαf)m‖L1,μ≤Csup⁡m≥0‖fm‖L1,λ,for allℒ1,λ-bounded martingales f=(fm)m≥0.To prove (5.39) we show that, for any n, any m>n and any Bn∈A(ℱn),(5.40)∫Bn|(Iαf)m-(Iαf)n|dP≲P(Bn)1+μ‖fm‖L1,λ.Note that(5.41)(Iαf)m-(Iαf)n=∑k=n+1mbk-1αdkf=∑k=n+1m-1(bk-1α-bkα)(fk-fn)+bm-1α(fm-fn).Sincebm is nonincreasing with respect to m, we have(5.42)∫Bnbm-1α|fm-fn|dP≤P(Bn)α∫Bn|fm-fn|dP≲P(Bn)1+μ‖fm‖L1,λ.Therefore, to prove (5.40), we only need to show that(5.43)∑k=n+1m∫Bn(bk-1α-bkα)|fk-fn|dP≤P(Bn)1+μ‖fm‖L1,λ∀m>n.Using the inequality|fk-fn|≤Ek|fm-fn| for n<k≤m, we have(5.44)∫Bn(bk-1α-bkα)|fk-fn|dP≤∫Bn(bk-1α-bkα)Ek|fm-fn|dP=∫Bn(bk-1α-bkα)|fm-fn|dP.Therefore, we have(5.45)∑k=n+1m∫Bn(bk-1α-bkα)|fk-fn|dP≤∑k=n+1m∫Bn(bk-1α-bkα)|fm-fn|dP=∫Bn(bnα-bmα)|fm-fn|dP≤P(Bn)α∫Bn|fm-fn|dP≤P(Bn)1+μ‖fm‖L1,λ.We have obtained (5.43). The proof of Theorem 5.8 is completed.


## 5.1. Boundedness of Fractional Integrals

The following is forLp-Lq boundedness.Theorem 5.1.
Assume that{ℱn}n≥0 is regular. Let  1≤p<q<∞,  -1/p+α=-1/q. Then, for  f∈L1,
(5.2)‖(Iαf)*‖Lq≤C‖f‖Lp,ifp>1,(5.3)‖(Iαf)*‖WLq≤C‖f‖L1,ifp=1.Remark 5.2.
Letf=(fn)n≥0 be an L1-bounded martingale. For each L1-function fm, m≥0, consider the corresponding martingale fm=(En[fm])n≥0=(fmin(n,m))n≥0. Then (Iαfm)n=(Iαf)n  for m≥n. Therefore, from (5.3) it follows that, for m≥n,
(5.4)‖(Iαf)n‖WLq=‖(Iαfm)n‖WLq≲‖fm‖L1≤sup⁡n≥0‖fn‖L1.
This shows that
(5.5)sup⁡n≥0‖(Iαf)n‖WLq≤Csup⁡n≥0‖fn‖L1.Remark 5.3.
Let a martingalef=(fn)n≥0 be L1-bounded. Since WLq⊂Lq1 and ∥(Iαf)n∥Lq1≤∥(Iαf)n∥WLq for 1<q1<q, from Remark 5.2 it follows that the martingale ((Iαf)n)n≥0isLq1-bounded and that it converges in Lq1. Denote this limit by Iαf.Then, Iαf=∑k=0∞bk-1αdkf and En[Iαf]=(Iαf)n.
Ifp>1, then the same observation as in Remark 5.2 with (5.2) shows that
(5.6)sup⁡n≥0‖(Iαf)n‖Lq≤Csup⁡n≥0‖fn‖Lp.
Hence we have the following corollary.Corollary 5.4.
Assume that{ℱn}n≥0 is regular. Let  1≤p<q<∞,  -1/p+α=-1/q. Then
(5.7)Iα∈B(Lp,Lq),ifp>1,Iα∈B(L1,WLq),ifp=1.For Morrey norms, one has the following.Theorem 5.5.
Assume that{ℱn}n≥0 is regular. Let  1≤p<∞,  -1/p≤λ<0, and  μ=λ+α<0. Then, for  f∈L1,
(5.8)‖(Iαf)*‖Lq,μ≤C‖f‖Lp,λ,ifp>1,1≤q≤(λ/μ)p,(5.9)‖(Iαf)*‖Lq,μ≤C‖f‖L1,λ,ifp=1,1≤q<(λ/μ)p,(5.10)‖(Iαf)*‖WLq,μ≤C‖f‖L1,λ,ifp=1,q=(λ/μ)p.Note that Theorem5.1 is not a corollary of Theorem 5.5, since ∥f∥Lp≲∥f∥Lp,-1/p≤∥f∥Lp in general.Remark 5.6.
In order to prove (5.8) it suffices to prove it in the case where q=(λ/μ)p, since ∥f∥Lq1,λ≤∥f∥Lq,λ for q1≤q by Hölder’s inequality. The inequality (5.9) follows from (5.10), since ∥f∥Lq1,λ≲∥f∥WLq,λ for q1<q.By the same observation as in Remark5.2 we have the following.Corollary 5.7.
Assume that{ℱn}n≥0 is regular. Let  1≤p<∞,  -1/p≤λ<0, and  μ=λ+α<0. Then
(5.11)Iα∈B(Lp,λ,Lq,μ),ifp>1,1≤q≤(λ/μ)p,Iα∈B(L1,λ,Lq,μ),ifp=1,1≤q<(λ/μ)p,Iα∈B(L1,λ,WLq,μ),ifp=1,q=(λ/μ)p.For Campanato spaces, one has the following.Theorem 5.8.
Assume that{ℱn}n≥0 is regular. Let  1≤p<∞,  λ≥-1/p,  1≤q<∞, and  μ=λ+α≥0. Then
(5.12)‖Iαf‖Lq,μ≤C‖f‖Lp,λ,f∈L1,
and
(5.13)Iα∈B(Lp,λ,Lq,μ),
where Iαf in (5.12) denotes the limit function described in Remark 5.3, for f=(Enf)n≥0, f∈L1.Remark 5.9.
Ifμ<0, then, by Theorem 3.1 (ii), the inequalities (5.8) and (5.9) in Theorem 5.5 hold with Lp,λ and Lq,μ replaced by ℒp,λ and ℒq,μ, respectively.
If1<p<∞ and -1/p=λ, then ∥f∥ℒ1,λ~∥f∥L1,λ≲∥f∥WLp≲∥f∥Lp by Remark 2.5 and Proposition 3.10. Since Lp⫋L1,λ in general, the following is an improvement of the results in [21].Corollary 5.10.
Assume that{ℱn}n≥0 is regular. Then, for  f∈L1,
(5.14)‖Iαf‖BMO≤C‖f‖L1,-1/p,if-1p+α=0,1<p<∞,‖Iαf‖Lipβ≤C‖f‖L1,-1/p,if-1p+α=β>0,1<p<∞,‖Iαf‖Lipα≤C‖f‖BMO,‖Iαf‖Lipγ≤C‖f‖Lipβ,ifβ+α=γ,β>0,(5.15)Iα∈B(L1,-1/p,BMO),if-1p+α=0,1<p<∞,Iα∈B(L1,-1/p,Lipβ),if-1p+α=β>0,1<p<∞,Iα∈B(BMO,Lipα),Iα∈B(Lipβ,Lipγ),ifβ+α=γ,β>0.


## 5.2. Proof of Theorem5.5

By Remark5.6 we only need to prove (5.8) and (5.10) in the case q=(λ/μ)p. So we always assume that q=(λ/μ)p in this proof.First we show the pointwise estimate(5.16)(Iαf)*(ω)≤Cf*(ω)μ/λ‖f‖Lp,λ-α/λ,whereC is a positive constant independent of ω. To do this we prove that, for any m≥1 and any Bm∈A(ℱm),(5.17)|(Iαf)m(ω)|≤Cf*(ω)μ/λ‖f‖Lp,λ-α/λ,ω∈Bm.TakeBk∈A(ℱk), 0≤k<m, such that Bm⊂Bm-1⊂…⊂B0 and let(5.18)K={k:0<k≤m,Bk≠Bk-1}={k1,k2,…,kl},where0=k0<k1<k2<…<kℓ. Then, by Lemma 3.3,(5.19)(1+1R)bkj≤bkj-1≤Rbkjon⁡Bm.Hence, we can write(5.20)(Iαf)m=∑0<kj≤mbkj-1αdkjf=∑j=1lbkj-1αdkjfon⁡Bm,sincebk=bk-1 and dkf=0 for k∉K. Note that for ω∈Bm,(5.21)|dkjf(ω)|=|fkj(ω)-fkj-1(ω)|≤|fkj(ω)|+|fkj-1(ω)|=|1P(Bkj)∫BkjfdP|+|1P(Bkj-1)∫Bkj-1fdP|≤(1P(Bkj)∫Bkj|f|pdP)1/p+(1P(Bkj-1)∫Bkj-1|f|pdP)1/p≤(P(Bkj)λ+P(Bkj-1)λ)‖f‖Lp,λ≤(1+(1R)λ)bkj-1(ω)λ‖f‖Lp,λ.Then, forω∈Bm and for 0<n≤m,(5.22)|∑k=0nbk-1(ω)αdkf(ω)|≤(1+(1R)λ)∑0<kj≤nbkj-1(ω)μ‖f‖Lp,λ≤1+(1/R)λ1-(1+1/R)μbn(ω)μ‖f‖Lp,λ.On the other hand, for0≤n<m, letting j(n)=min{j:n<kj}, we have, for ω∈Bm,(5.23)∑k=n+1mbk-1(ω)αdkf(ω)=∑j=j(n)lbkj-1(ω)αdkjf(ω)=∑j=j(n)lbkj-1(ω)αfkj(ω)-∑j=j(n)lbkj-1(ω)αfkj-1(ω)=bkl-1(ω)αfkl(ω)+∑j=j(n)l-1(bkj-1(ω)α-bkj(ω)α)fkj(ω)-bkj(n)-1(ω)αfkj(n)-1(ω),(5.24)|∑k=n+1mbk-1(ω)αdkf(ω)|≤bkl-1(ω)αf*(ω)+∑j=j(n)l-1|bkj-1(ω)α-bkj(ω)α|f*(ω)+bkj(n)-1(ω)αf*(ω)≤2bkj(n)-1(ω)αf*(ω)=2bn(ω)αf*(ω).Here, let(5.25)Ω1={ω∈Ω:(f*(ω)‖f‖Lp,λ)1/λ≤b0(ω)},Ω2=Ω∖Ω1.Ifω∈Ω1∩Bm and(5.26)(f*(ω)‖f‖Lp,λ)1/λ≤bm(ω),then, by (5.22) we have(5.27)|(Iαf)m(ω)|≲bm(ω)μ‖f‖Lp,λ≤(f*(ω)‖f‖Lp,λ)μ/λ‖f‖Lp,λ=f*(ω)μ/λ‖f‖Lp,λ-α/λ,sinceμ<0 and 1-μ/λ=-α/λ. If ω∈Ω1∩Bm and(5.28)bm(ω)<(f*(ω)‖f‖Lp,λ)1/λ,choosingn such that(5.29)R-1bn(ω)<(f*(ω)‖f‖Lp,λ)1/λ≤bn(ω),we have by (5.22) and (5.24)(5.30)|(Iαf)m(ω)|≲bn(ω)μ‖f‖Lp,λ+bn(ω)αf*(ω)≲(f*(ω)‖f‖Lp,λ)μ/λ∥f∥Lp,λ+(f*(ω)‖f‖Lp,λ)α/λf*(ω)≲f*(ω)μ/λ‖f‖Lp,λ-α/λ.Ifω∈Ω2∩Bm, then by (5.24) we have(5.31)|(Iαf)m(ω)|=|∑k=1mbk-1(ω)αdkf(ω)|≲b0(ω)αf*(ω)≤(f*(ω)‖f‖Lp,λ)α/λf*(ω)=f*(ω)μ/λ‖f‖Lp,λ-α/λ.Therefore, we have (5.17).Next, applying the boundedness of the maximal function (Theorem4.1), we show(5.32)(1P(B)∫B[(Iαf(ω))*]qdP)1/q≲P(B)μ‖f‖Lp,λ,ifp>1,(5.33)sup⁡t>0t(P(B∩{(Iαf)*>t})P(B))1/q≲P(B)μ‖f‖L1,λ,ifp=1,for anyn and any B∈A(ℱn). To show (5.32), using (5.16) and the relations qμ/λ=p and -α/λ=1-p/q, we have(5.34)(1P(B)∫B[(Iαf(ω))*]qdP)1/q≲(1P(B)∫Bf*(ω)qμ/λdP)1/q‖f‖Lp,λ-α/λ=(1P(B)∫Bf*(ω)pdP)(1/p)(p/q)‖f‖Lp,λ1-p/q.Applying the boundedness of the maximal function, we have(5.35)(1P(B)∫Bf*(ω)pdP)(1/p)(p/q)≤(P(B)λ‖f*‖Lp,λ)p/q≲P(B)μ‖f‖Lp,λp/q.Then we have (5.32). To show (5.33), using (5.16) and the relations p=1, μ/λ=1/q and -α/λ=1-1/q, we have(5.36)sup⁡t>0t(P(B∩{(Iαf)*>t})P(B))1/q≲sup⁡t>0t(P(B∩{(f*)μ/λ‖f‖L1,λ-α/λ>t})P(B))1/q=sup⁡t>0t(P(B∩{(f*)1/q∥f∥L1,λ1-1/q>t})P(B))1/q=sup⁡t>0t(P(B∩{(f*)1/q>t})P(B))1/q‖f‖L1,λ1-1/q=(sup⁡t>0tP(B∩{(f*)>t})P(B))1/q‖f‖L1,λ1-1/q.Applying the boundedness of the maximal function, we have(5.37)(sup⁡t>0tP(B∩{(f*)>t})P(B))1/q≤(P(B)λ‖f*‖WL1,λ)1/q≲P(B)μ‖f‖Lp,λ1/q.Then we have (5.33). The proof is complete.


## 5.3. Proof of Theorem5.1

Letλ=-1/p and μ=-1/q. Then μ/λ=p/q and -α/λ=(1/p-1/q)p=1-p/q. In this case the pointwise estimate (5.16) implies(5.38)(Iαf)*(ω)≤Cf*(ω)p/q‖f‖Lp,-1/p1-p/q≤Cf*(ω)p/q‖f‖Lp1-p/q.Applying the boundedness of the maximal function onLp, we get the conclusion.


## 5.4. Proof of Theorem5.8

Note that, forf∈L1, ∥f∥ℒp,λ=∥f-E0f∥ℒp,λ and Iαf=Iα(f-E0f). Then it is sufficient to prove (5.12) only for f∈L10. Therefore, (5.12) follows from (5.13) and Proposition 2.2(ii).By Hölder’s inequality we have∥f∥ℒ1,λ≤∥f∥ℒp,λ for p≥1. Furthermore, by Theorem 3.1(iii) and (iv) we have ∥Iαf∥ℒq,μ~∥Iαf∥ℒ1,μ for q≥1. So we only need to prove that(5.39)sup⁡m≥0‖(Iαf)m‖L1,μ≤Csup⁡m≥0‖fm‖L1,λ,for allℒ1,λ-bounded martingales f=(fm)m≥0.To prove (5.39) we show that, for any n, any m>n and any Bn∈A(ℱn),(5.40)∫Bn|(Iαf)m-(Iαf)n|dP≲P(Bn)1+μ‖fm‖L1,λ.Note that(5.41)(Iαf)m-(Iαf)n=∑k=n+1mbk-1αdkf=∑k=n+1m-1(bk-1α-bkα)(fk-fn)+bm-1α(fm-fn).Sincebm is nonincreasing with respect to m, we have(5.42)∫Bnbm-1α|fm-fn|dP≤P(Bn)α∫Bn|fm-fn|dP≲P(Bn)1+μ‖fm‖L1,λ.Therefore, to prove (5.40), we only need to show that(5.43)∑k=n+1m∫Bn(bk-1α-bkα)|fk-fn|dP≤P(Bn)1+μ‖fm‖L1,λ∀m>n.Using the inequality|fk-fn|≤Ek|fm-fn| for n<k≤m, we have(5.44)∫Bn(bk-1α-bkα)|fk-fn|dP≤∫Bn(bk-1α-bkα)Ek|fm-fn|dP=∫Bn(bk-1α-bkα)|fm-fn|dP.Therefore, we have(5.45)∑k=n+1m∫Bn(bk-1α-bkα)|fk-fn|dP≤∑k=n+1m∫Bn(bk-1α-bkα)|fm-fn|dP=∫Bn(bnα-bmα)|fm-fn|dP≤P(Bn)α∫Bn|fm-fn|dP≤P(Bn)1+μ‖fm‖L1,λ.We have obtained (5.43). The proof of Theorem 5.8 is completed.



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*Source: 673929-2012-07-09.xml*