# Martingale Morrey-Hardy and Campanato-Hardy Spaces

Authors: Eiichi Nakai; Gaku Sadasue; Yoshihiro Sawano
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We introduce generalized Morrey-Campanato spaces of martingales,
which generalize both martingale Lipschitz spaces introduced by Weisz (1990) and martingale Morrey-Campanato spaces introduced in 2012. We also introduce generalized Morrey-Hardy and Campanato-Hardy spaces of martingales and study Burkholder-type equivalence. We give some results on the boundedness of fractional integrals of martingales on these spaces.


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## Body

## 1. Introduction

Lebesgue spacesLp and Hardy spaces Hp play an important role in martingale theory and in harmonic analysis as well. Morrey-Campanato spaces are very useful to know more precise properties of functions and martingales. It is known that Morrey-Campanato spaces contain Lp, BMO, and Lipα as special cases; see, for example, [1, 2].In martingale theory, Weisz [3] introduced martingale Lipschitz spaces for general filtrations {ℱn}n≥0 and proved the duality between martingale Hardy spaces and martingale Lipschitz spaces. This result was extended to generalized martingale Campanato spaces and martingale Orlicz-Hardy spaces in [4]. Recently, martingale Morrey-Campanato spaces were introduced in [5], where each sub-σ-algebra ℱn is generated by countable atoms.In this paper, we introduce martingale Morrey-Hardy and Campanato-Hardy spaces based on square functions and unify Hardy, Lipschitz, and Morrey-Campanato spaces in [3–5]. To do this, we first introduce generalized martingale Morrey-Campanato spaces by using subfamilies {ℬn}n≥0 of the filtration {ℱn}n≥0 with ℬn⊂ℱn for each n≥0. We establish Burkholder-type equivalence and discuss equivalence between martingale Morrey spaces and martingale Campanato spaces in a suitable condition. We also establish a John-Nirenberg-type theorem for generalized martingale Campanato-Hardy spaces; see Theorem 15.On these martingale spaces, we introduce generalized fractional integrals as martingale transforms and prove their boundedness. Our result extends several results in [5–7] to these spaces. The fractional integrals are very useful tools to analyse function spaces in harmonic analysis. Actually, on the Euclidean space, Hardy and Littlewood [8, 9] and Sobolev [10] investigated the fractional integrals to establish the theory of Lebesgue spaces and Lipschitz spaces. Stein and Weiss [11], Taibleson and Weiss [12], and Krantz [13] also investigated the fractional integrals to establish the theory of Hardy spaces; see also [14]. The Lp-Lq boundedness of the fractional integrals is well known as the Hardy-Littlewood-Sobolev theorem derived from [8–10]. This boundedness has been extended to Morrey-Campanato spaces by Peetre [1] and Adams [15]; see also Chiarenza and Frasca [16]. In martingale theory, based on the result on the Walsh multiplier by Watari [7, Theorem 1.1], Chao and Ombe [6] proved the boundedness of the fractional integrals for Hp, Lp, BMO, and Lipschitz spaces of the dyadic martingale. The boundedness of the fractional integrals for martingale Morrey-Campanato spaces was established in [5]. For other types of operators for martingales, see the recent work by Tanaka and Terasawa [17].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, are 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. Definitions and Notation

Let(Ω,Σ,P) be a probability space and ℱ={ℱn}n≥0 a nondecreasing sequence of sub-σ-algebras of Σ such that Σ=σ(⋃n‍ℱn). For the sake of simplicity, let ℱ-1=ℱ0. The set B∈ℱn is called atom, more precisely (ℱn,P)-atom, if any A⊂B, A∈ℱn, satisfying 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 toℱn are denoted by E and En, respectively. It is known from the Doob theorem that if p∈(1,∞), then any Lp-bounded martingale converges in Lp. Moreover, if p∈[1,∞), then, for any f∈Lp, its corresponding martingale (fn)n≥0 with fn=Enf is an Lp-bounded martingale and converges to f in Lp (see, e.g., [18]). For this reason a function f∈L1 and the corresponding martingale (fn)n≥0 will be denoted by the same symbol f.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, its corresponding martingale (fn)n≥0 with fn=Enf is an Lp-bounded martingale in ℳ. For this reason, we regard Lp0 as a subset of ℳ.Letℬ={ℬn}n≥0 be subfamilies of ℱ={ℱn}n≥0 with ℬn⊂ℱn for each n≥0. We denote by ℬ⊂ℱ this relation of ℬ and ℱ.In this paper, we always postulate the following condition onℬ(1)Thereexistsacountablesubsetℬ0′⊂ℬ0suchthatP(⋃B∈ℬ0′‍B)=1.
We first define generalized martingale Morrey-Campanato spaces with respect to ℬ as follows.Definition 1.
Letℬ⊂ℱ, p∈[1,∞), and ϕ:(0,1]→(0,∞). For f∈L1, let
(2)∥f∥Lp,ϕ=∥f∥Lp,ϕ(ℬ)=supn≥0supB∈ℬn1ϕ(P(B))(1P(B)∫B‍|f|pdP)1/p,∥f∥ℒp,ϕ=∥f∥ℒp,ϕ(ℬ)=supn≥0supB∈ℬn1ϕ(P(B))(1P(B)∫B‍|f-Enf|pdP)1/p,∥f∥ℒp,ϕ-=∥f∥ℒp,ϕ-(ℬ)=supn≥0supB∈ℬn1ϕ(P(B))(1P(B)∫B‍|f-En-1f|pdP)1/p,
and define
(3)Lp,ϕ=Lp,ϕ(ℬ)={f∈Lp0:∥f∥Lp,ϕ<∞},ℒp,ϕ=ℒp,ϕ(ℬ)={f∈Lp0:∥f∥ℒp,ϕ<∞},ℒp,ϕ-=ℒp,ϕ-(ℬ)={f∈Lp0:∥f∥ℒp,ϕ-<∞}.Remark 2.
By the condition (1), the functionals ∥f∥Lp,ϕ, ∥f∥ℒp,ϕ, and ∥f∥ℒp,ϕ- are norms on Lp0.Remark 3.
Letf∈L10. Then, f∈ℒp,ϕ if and only if its corresponding martingale (fn)n≥0 is ℒp,ϕ-bounded; that is, supn≥0∥fn∥ℒp,ϕ<∞. The same conclusion holds for ℒp,ϕ-. Furthermore, if each sub-σ-algebra ℱn is generated by countable atoms, ℬ={A(ℱn)}n≥0 and ϕ is almost decreasing, then the same conclusion holds for Lp,ϕ. More precisely, see Proposition 8.Remark 4.
In general,∥f∥ℒp,ϕ≤2∥f∥Lp,ϕ and hence Lp,ϕ⊂ℒp,ϕ. Actually, for any B∈ℬn,
(4)(∫B‍|f-Enf|pdP)1/p≤(∫B‍|f|pdP)1/p+(∫B‍|Enf|pdP)1/p≤2(∫B‍|f|pdP)1/p.
Similarly, ∥f∥ℒp,ϕ≤2∥f∥ℒp,ϕ- and ℒp,ϕ-⊂ℒp,ϕ.Definition 5.
Forϕ≡1, denote ℒp,ϕ and ℒp,ϕ- by BMOp and BMOp-, respectively. For ϕ(r)=rα, α>0, denote ℒp,ϕ and ℒp,ϕ- by Lipp,α and Lipp,α-, respectively.Ifϕ(r)=rλ,  λ∈(-∞,∞), then we simply denote Lp,ϕ, ℒp,ϕ, and ℒp,ϕ- by Lp,λ, ℒp,λ, and ℒp,λ-, respectively.A functionϕ:(0,1]→(0,∞) is said to be almost increasing (resp., almost decreasing) if there exists a positive constant C such that
(5)ϕ(s)≤Cϕ(t)(resp.,ϕ(t)≤Cϕ(s))for0<s≤t≤1.For the caseℬ=ℱ, the spaces BMOp(ℱ) and Lipp,α(ℱ) with α≥0, were introduced by Weisz [3].Recall thatA(ℱn) is the set of all atoms in ℱn and let 𝒜={A(ℱn)}n≥0. Suppose that each sub-σ-algebra ℱn is generated by countable atoms for the time being. Then, BMOp(ℱ)=BMOp(𝒜) and Lipp,α(ℱ)=Lipp,α(𝒜); see [5]. In general, if ϕ is almost increasing, then
(6)Lp,ϕ(ℱ)=Lp,ϕ(𝒜),ℒp,ϕ(ℱ)=ℒp,ϕ(𝒜),ℒp,ϕ-(ℱ)=ℒp,ϕ-(𝒜),with equivalent norms, respectively. However, if ϕ is not almost increasing, then these equalities fail in general; see [5].In this paper, we do not always assume that each sub-σ-algebra ℱn is generated by countable atoms. Let
(7)A(ℱn)⊥={B∈ℱn:P(B∩A)=0∀A∈A(ℱn)},
and let
(8)ℬn=A(ℱn)∪A(ℱn)⊥(n≥0).
In this case, if ϕ is almost increasing, then we will show that
(9)Lp,ϕ(ℱ)=Lp,ϕ(ℬ),ℒp,ϕ(ℱ)=ℒp,ϕ(ℬ),ℒp,ϕ-(ℱ)=ℒp,ϕ-(ℬ),
with equivalent norms, respectively (see Proposition 9). Moreover, if ℱ0 is nonatomic, then ℬn=ℱn for all n≥0. If each sub-σ-algebra ℱn is generated by countable atoms, then ℬn=A(ℱn) for all n≥0. Therefore, our definition generalizes those in [3–5].Next we define Morrey-Hardy and Campanato-Hardy spaces, based on square functions, with respect toℬ as follows. For f∈ℳ, we denote by Sn(f) and S(f) the square function of f:
(10)Sn(f)=(∑k=0n‍|dkf|2)1/2,S(f)=(∑k=0∞‍|dkf|2)1/2,
where dkf=fk-fk-1 (n≥0, with convention d0f=0 and S-1(f)=0). We further define
(11)S(n)(f)=(S(f)2-Sn(f)2)1/2=(∑k=n+1∞‍|dkf|2)1/2.Definition 6.
Letℬ⊂ℱ, p∈(0,∞), and ϕ:(0,1]→(0,∞). For f=(fn)n≥0∈ℳ, let
(12)∥f∥Hp,ϕS=∥f∥Hp,ϕS(ℬ)=supn≥0supB∈ℬn1ϕ(P(B))(1P(B)∫B‍S(f)pdP)1/p,∥f∥ℋp,ϕS=∥f∥ℋp,ϕS(ℬ)=supn≥0supB∈ℬn1ϕ(P(B))(1P(B)∫B‍S(n)(f)pdP)1/p,∥f∥ℋp,ϕS-=∥f∥ℋp,ϕS-(ℬ)=supn≥0supB∈ℬn1ϕ(P(B))(1P(B)∫B‍S(n-1)(f)pdP)1/p,
and define
(13)Hp,ϕS=Hp,ϕS(ℬ)={f∈ℳ:∥f∥Hp,ϕS<∞},ℋp,ϕS=ℋp,ϕS(ℬ)={f∈ℳ:∥f∥ℋp,ϕS<∞},ℋp,ϕS-=ℋp,ϕS-(ℬ)={f∈ℳ:∥f∥ℋp,ϕS-<∞}.By (1), the functionals ∥f∥Hp,ϕS, ∥f∥ℋp,ϕS, and ∥f∥ℋp,ϕS- are quasinorms on ℳ.Remark 7.
If we takeϕ≡1 and ℬ=ℱ, then the norm ∥f∥ℋp,ϕS- coincides with the norm ∥f∥BMOpS in [19, Definition 2.45]. In this point, our notation is different from the one in [19].In the end of this section, we present the definition of regularity onℱ and the doubling condition on ϕ. The filtration ℱ={ℱn}n≥0 is said to be regular, if there exists a constant R≥2 such that
(14)fn≤Rfn-1
holds for all nonnegative martingales (fn)n≥0. We say the smallest constant R satisfying (14) the regularity constant of ℱ. A function ϕ:(0,1]→(0,∞) is said to satisfy the doubling condition if there exists a positive constant Cϕ such that
(15)Cϕ-1≤ϕ(s)ϕ(t)≤Cϕ∀s,t∈(0,1]with12≤st≤2.
The smallest constant Cϕ satisfying (15) is called the doubling constant of ϕ.


## 3. Properties of Morrey-Hardy and Campanato-Hardy Spaces

In this section, we investigate the properties of Morrey-Hardy and Campanato-Hardy spaces. The proofs of the results in this section will be given in Section6.First we state basic properties of the norms.Proposition 8.
Letℬ⊂ℱ, p∈[1,∞) and ϕ:(0,1]→(0,∞). Let f∈L1 and let (fn)n≥0 be its corresponding martingale; fn=Enf. Then
(16)∥f∥Lp,ϕ≤supn≥0∥fn∥Lp,ϕ,∥f∥ℒp,ϕ=supn≥0∥fn∥ℒp,ϕ,∥f∥ℒp,ϕ-=supn≥0∥fn∥ℒp,ϕ-.
Moreover, if each sub-σ-algebra ℱn is generated by countable atoms, ℬ=𝒜, and ϕ is almost decreasing; that is, there exists a positive constant C0, such that ϕ(t)≤C0ϕ(s) for 0<s≤t≤1, then
(17)∥f∥Lp,ϕ≤supn≥0∥fn∥Lp,ϕ≤C0∥f∥Lp,ϕ.Proposition 9.
LetA(ℱn)∪A(ℱn)⊥⊂ℬn⊂ℱn(n≥0). If ϕ is almost increasing; that is, there exists a positive constant C0, such that ϕ(s)≤C0ϕ(t) for 0<s≤t≤1, then
(18)∥f∥Lp,ϕ(ℬ)≤∥f∥Lp,ϕ(ℱ)≤C0∥f∥Lp,ϕ(ℬ),
and the same conclusions hold for ∥·∥ℒp,ϕ, ∥·∥ℒp,ϕ-, ∥·∥Hp,ϕS, ∥·∥ℋp,ϕS, and ∥·∥ℋp,ϕS-. Consequently,
(19)Lp,ϕ(ℱ)=Lp,ϕ(ℬ),ℒp,ϕ(ℱ)=ℒp,ϕ(ℬ),ℒp,ϕ-(ℱ)=ℒp,ϕ-(ℬ),Hp,ϕ(ℱ)=Hp,ϕ(ℬ),ℋp,ϕS(ℱ)=ℋp,ϕS(ℬ),ℋp,ϕS-(ℱ)=ℋp,ϕS-(ℬ),
with equivalent norms, respectively.Forp∈(0,∞), let HpS be the set of all f∈ℳ such that ∥S(f)∥Lp<∞. Let ∥f∥HpS=∥S(f)∥Lp. Note that if ϕ(r)=r-1/p and Ω∈ℬ0, then Hp,ϕS=HpS and ∥f∥Hp,ϕS=∥f∥HpS.The following is well known as Burkholder’s inequality.Theorem 10 (Burkholder [20]).
Ifp∈(1,∞), then there exist positive constants cp and Cp, that depend only on p, such that
(20)cp∥f∥Lp≤∥f∥HpS≤Cp∥f∥Lp
for all f∈L10⊂ℳ.For expressions of the constantscp and Cp, see, for example, [21–23]. See also [24] for Burkholder’s inequality on Banach functions spaces.Our first result is the following, which is an extension of Burkholder’s inequality to martingale Campanato spaces.Theorem 11.
Letℬ⊂ℱ, p∈(1,∞) and ϕ:(0,1]→(0,∞). Then
(21)cp∥f∥ℒp,ϕ≤∥f∥ℋp,ϕS≤Cp∥f∥ℒp,ϕ,(22)cp1+cp∥f∥ℒp,ϕ-≤∥f∥ℋp,ϕS-≤(2Cp+1)∥f∥ℒp,ϕ-
for all f∈L10⊂ℳ, where cp and Cp are the constants in Theorem 10.Next we give the relations betweenℒp,ϕ and ℒp,ϕ- and between ℋp,ϕS and ℋp,ϕS-. We consider the following condition on ℬ:
(23){ω∈Ω:En-1[χB](ω)>0}∈ℬn-1∀B∈ℬn(n≥1).Theorem 12.
Letℬ⊂ℱ and ϕ:(0,1]→(0,∞). Then
(24)ℒp,ϕ⊃ℒp,ϕ-with∥f∥ℒp,ϕ≤2∥f∥ℒp,ϕ-forp∈[1,∞),(25)ℋp,ϕS⊃ℋp,ϕS-with∥f∥ℋp,ϕS≤∥f∥ℋp,ϕS-forp∈(0,∞).
Conversely, if ℱ is regular, ℬ satisfies (23), and ϕ satisfies the doubling condition, then there exists a positive constant C, dependent only on p, the regularity constant of ℱ, and the doubling constant of ϕ, such that
(26)ℒp,ϕ⊂ℒp,ϕ-withC∥f∥ℒp,ϕ≥∥f∥ℒp,ϕ-forp∈[1,∞),(27)ℋp,ϕS⊂ℋp,ϕS-withC∥f∥ℋp,ϕS≥∥f∥ℋp,ϕS-forp∈(0,∞).We give a relation between martingale Morrey spaces and martingale Campanato spaces in the following form.Theorem 13.
Suppose that everyσ-algebra ℱn is generated by countable atoms. Let ℬ=𝒜, p∈[1,∞), and ϕ:(0,1]→(0,∞). Assume that ϕ satisfies the doubling condition and there exists a positive constant Cϕ′ such that
(28)∫r1‍ϕ(t)tdt≤Cϕ′ϕ(r)(0<r<1).
Then, there exists a positive constant C such that
(29)12∥f∥ℒp,ϕ≤∥f∥Lp,ϕ≤C∥f∥ℒp,ϕ-∀f∈L10,∥f∥ℋp,ϕS-≤∥f∥Hp,ϕS≤C∥f∥ℋp,ϕS-∀f∈ℳ.
Moreover, if ℱ is regular, then ∥f∥Lp,ϕ, ∥f∥ℒp,ϕ, and ∥f∥ℒp,ϕ- are equivalent to each other, and ∥f∥Hp,ϕS, ∥f∥ℋp,ϕS and ∥f∥ℋp,ϕS- are equivalent to each other.Using Theorems11–13, we have Burkholder-type equivalence for generalized martingale Morrey spaces.Corollary 14.
Suppose that everyσ-algebra ℱn is generated by countable atoms. Let ℬ=𝒜, p∈(1,∞), and ϕ:(0,1]→(0,∞). Assume that ϕ satisfies the doubling condition and there exists a positive constant Cϕ′ such that
(30)∫r1‍ϕ(t)tdt≤Cϕ′ϕ(r)(0<r<1).
If ℱ is regular, then there exist positive constants c and C such that, for all f∈L10,
(31)c∥f∥Lp,ϕ≤∥f∥Hp,ϕS≤C∥f∥Lp,ϕ.For the martingaleBMO spaces based on square functions, the John-Nirenberg-type equivalence was established by Weisz [25] and [19, Theorem 2.50]. We extend this theorem to the spaces ℋp,ϕS and ℋp,ϕS-.Theorem 15.
LetA(ℱn)∪A(ℱn)⊥⊂ℬn⊂ℱn(n≥0), p∈(0,∞) and ϕ:(0,1]→(0,∞). Assume that ϕ is almost increasing and satisfies the doubling condition. Then, ∥f∥ℋp,ϕS≤Cp,q,ϕ∥f∥ℋq,ϕS- for all q∈(0,∞). If we further assume that ℱ is regular, then ∥f∥ℋp,ϕS and ∥f∥ℋp,ϕS- are equivalent to ∥f∥ℋ2,ϕS.


## 4. Fractional Integrals

In this section, we state the results on the boundedness of fractional integrals as martingale transforms. The proofs of the results in this section will be given in Section7.Let(γn)n≥0 be a sequence of nonnegative bounded functions adapted to ℱ={ℱn}n≥0; that is, γn is ℱn-measurable for every n≥0. Let Iγ be the martingale transform associate to (γn)n≥0; that is,
(32)(Iγf)n=∑k=0n‍γk-1dkf,
with convention γ-1d0f=0. Note that if f=(fn)n≥0∈ℳ, then Iγf=((Iγf)n)n≥0∈ℳ.We now define a generalized fractional integralIρ for martingales as a special case of Iγ under the assumption that every σ-algebra ℱn is generated by countable atoms. Our definition generalizes the fractional integral for dyadic martingales introduced in [6, 7]. The idea of Iρ comes from [26].Suppose that everyσ-algebra ℱn is generated by countable atoms. Let bn be an ℱn-measurable function such that
(33)bn(ω)=P(B)fora.s.ω∈BwithB∈A(ℱn);
that is,
(34)bn=∑B∈A(ℱn)‍P(B)χBa.s.
For a bounded function ρ:(0,1]→(0,∞), we define a generalized fractional integral Iρf=((Iρf)n)n≥0 of f=(fn)n≥0∈ℳ by
(35)(Iρf)n=∑k=0n‍ρ(bk-1)dkf.
The generalized fractional integral Iρ is obtained by taking γn=ρ(bn) in (32). If ρ(r)=rα,α>0, then we simply denote Iρ by Iα.For quasinormed spacesM1 and M2 of martingales, we denote by B(M1,M2) the set of all bounded martingale transforms from M1 to M2; that is, T∈B(M1,M2) means that there exists a positive constant C such that
(36)∥Tf∥M2≤C∥f∥M1
for all martingales f=(fn)n≥0∈M1.We first study the boundedness on the spacesℋp,ϕS. On martingale Campanato-Hardy spaces, we consider the fractional integral as a martingale transform associated with monotone multipliers. We say a sequence of nonnegative measurable functions γ=(γn)n≥0 is almost decreasing if there exists a positive constant C such that
(37)γk(ω)≤Cγℓ(ω)a.s.∀k≥ℓ.
For an almost decreasing sequence γ=(γn)n≥0, we define Aγ by
(38)Aγ=inf{C>0:Csatisfies(37)}.In Theorem16 below, we do not need any assumption on {ℱn}n≥0.Theorem 16.
Letℬ⊂ℱ, p∈(0,∞), and ϕ,ψ:(0,1]→(0,∞). Let (γn)n≥0 be a sequence of nonnegative bounded almost decreasing adapted functions, and let Iγ be the martingale transform defined by (32). Assume that
(39)Cγ,ϕ,ψ=supn≥0supB∈ℬnϕ(P(B))ψ(P(B))∥γnχB∥L∞<∞.
Then
(40)Iγ∈B(ℋp,ϕS,ℋp,ψS)
with
(41)∥Iγf∥ℋp,ψS≤AγCγ,ϕ,ψ∥f∥ℋp,ϕS.If everyσ-algebra ℱn is generated by countable atoms, then we can apply Theorem 16 to the generalized fractional integral Iρ. The following corollary extends [5, Theorem 5.8] to the spaces ℋp,ϕS.Corollary 17.
Assume that everyσ-algebra ℱn is generated by countable atoms and ℬ=𝒜. Let p∈(0,∞) and ρ,ϕ,ψ:(0,1]→(0,∞). Suppose that ρ is almost increasing and that
(42)sup0<t≤1ρ(t)ϕ(t)ψ(t)<∞.
Then
(43)Iρ∈B(ℋp,ϕS,ℋp,ψS).
If one further assumes that {ℱn}n≥0 is regular and that ψ is almost increasing and satisfies the doubling condition, then
(44)Iρ∈B(ℋp,ϕS,ℋq,ψS),(p,q∈(0,∞)).We next study the boundedness on martingale Morrey-Hardy spacesHp,ϕS and martingale Hardy spaces HpS.Recall thatA(ℱn)⊥=ℱn for all n≥0 if ℱ0 is nonatomic.Proposition 18.
Letℬ⊂ℱ, 0<p<q<∞, and ϕ:(0,1]→(0,∞). Let (γn)n≥0 be a sequence of adapted functions. Suppose that ℱ0 is nonatomic and that ℬ=ℱ. Assume in addition that ϕ is almost decreasing, that t1/pϕ(t) is almost increasing, and that limt→0ϕ(t)=∞. Then, Iγ∉B(Hp,ϕS,Hq,ϕp/qS)∖{0}.According to Proposition18, to consider the boundedness on Hp,ϕS and HpS, we suppose that every σ-algebra ℱn is generated by countable atoms and that ℬ=𝒜.In this case, ifϕ(r)=r-1/p and ℱ0={Ω,∅}, then Hp,ϕS coincides with HpS and ∥f∥Hp,ϕS=∥f∥HpS. However, if ℱ0≠{Ω,∅}, then Hp,ϕS does not coincide with HpS in general. We do not always assume that ℱ0={Ω,∅}.Theorem 19.
Suppose that everyσ-algebra ℱn is generated by countable atoms, that ℬ=𝒜, and that {ℱn}n≥0 is regular. Let 0<p<q<∞ and ϕ:(0,1]→(0,∞), and let (γn)n≥0 be a sequence of nonnegative bounded adapted functions. Assume that ϕ satisfies the doubling condition and that there exists a positive constant C such that
(45)∑k=0n‍γk-1ϕ(bk-1)χ{bk≠bk-1}+ϕ(bn)×∑k=n+1∞‍γk-1χ{bk≠bk-1}≤Cϕ(bn)p/qa.s.
for all n≥0, where bk is the measurable function defined by (33). Then
(46)Iγ∈B(Hp,ϕS,Hq,ϕp/qS).
Furthermore, if ϕ(t)=t-1/p, then
(47)Iγ∈B(HpS,HqS).As a consequence of Theorem19, we have the following corollary, which gives an extension of [5, Corollary 5.7] to the spaces Hp,ϕS and gives a martingale Morrey-Hardy version of Gunawan [27, Theorem B]:Corollary 20.
Suppose that everyσ-algebra ℱn is generated by countable atoms, that ℬ=𝒜, and that {ℱn}n≥0 is regular. Let 0<p<q<∞ and ρ,ϕ:(0,1]→(0,∞). Assume that ρ is bounded, that both ρ and ϕ satisfy the doubling condition, and that there exists a positive constant C such that
(48)ϕ(r)∫0r‍ρ(t)tdt+∫r1‍ϕ(t)ρ(t)tdt≤Cϕ(r)p/qϕ(r)∫0r‍ρ(t)tdt+∫r1‍ϕ(t)ρ(t)t5(0<r<1).
Then
(49)Iρ∈B(Hp,ϕS,Hq,ϕp/qS).The following extends the results for dyadic martingales in [6, 7] and the result for 0<p≤1 in [28].Corollary 21.
Suppose that everyσ-algebra ℱn is generated by countable atoms, that ℬ=𝒜, and that {ℱn}n≥0 is regular. Let 0<p<q<∞ and -1/p+α=-1/q. Then
(50)Iα∈B(HpS,HqS).


## 5. Lemmas

We prepare some lemmas to prove the results in Sections3 and 4.Lemma 1.
Letℬn satisfy A(ℱn)∪A(ℱn)⊥⊂ℬn⊂ℱn(n≥0). Suppose that ϕ:(0,1]→(0,∞) is almost increasing; that is, ϕ(r)≤C0ϕ(s) for all 0<r≤s≤1. Then, for all nonnegative functions F,
(51)supB∈ℱn1ϕ(P(B))(1P(B)∫B‍FdP)≤C0supB∈ℬn1ϕ(P(B))(1P(B)∫B‍FdP).Proof.
Let(52)𝒩=supB∈ℬn1ϕ(P(B))(1P(B)∫B‍FdP).
For any B∈ℱn, we can choose the sets Bj, j=0,1,2,… (finite or infinite) such that
(53)B=∪jBj,B0∈A(ℱn)⊥,Bj∈A(ℱn),B=∪jBj,B0∈A(ℱn)⊥,.j=1,2,…,P(B)=∑j‍P(Bj).
In this case, Bj∈ℬn, j=0,1,2,…, since A(ℱn)∪A(ℱn)⊥⊂ℬn. Then
(54)1ϕ(P(B))(1P(B)∫B‍FdP)=1ϕ(P(B))1P(B)∑j‍(∫Bj‍FdP)=∑j‍P(Bj)P(B)(1ϕ(P(B))P(Bj)∫Bj‍FdP)≤∑j‍P(Bj)P(B)(C0ϕ(P(Bj))P(Bj)∫Bj‍FdP)≤C0∑j‍P(Bj)P(B)𝒩=C0𝒩.
This shows the conclusion.Lemma 2 (see [5, Lemma 3.3]).
Suppose that everyσ-algebra ℱn is generated by countable atoms and that {ℱn}n≥0 is regular. Then, every sequence
(55)B0⊃B1⊃⋯⊃Bn⊃⋯,Bn∈A(ℱn),
has the following property: for each n≥1,
(56)Bn=Bn-1or(1+1R)P(Bn)≤P(Bn-1)≤RP(Bn),
where R is the constant in (14).Lemma 3.
Suppose that everyσ-algebra ℱn is generated by countable atoms and that {ℱn}n≥0 is regular. For B∈A(ℱm), let Bj∈A(ℱj) be
(57)B=Bm⊂Bm-1⊂⋯⊂B0.
Letϕ:(0,1]→(0,∞). Suppose that ϕ satisfies the doubling condition. Then, there exists a positive constant C, that depends only on ϕ and the regularity constant R, such that
(58)∑j=ℓ+1m‍ϕ(bj)χ{bj≠bj-1}≤C∫bmbℓ‍ϕ(t)tdtonB,
where bj is the function defined by (33).Proof.
LetJ={j:bj≠bj-1}. Then, by Lemma 2, we have
(59)∑j=ℓ+1m‍ϕ(bj)χ{bj≠bj-1}=∑j∈J,ℓ<j≤m‍ϕ(bj)=∑j∈J,ℓ<j≤m‍1log(bj-1/bj)∫bjbj-1‍ϕ(bj)tdt≲∑j∈J,ℓ<j≤m‍∫bjbj-1‍ϕ(t)tdt=∫bmbℓ‍ϕ(t)tdt.In Theorem13, we do not assume that {ℱn}n≥0 is regular. Hence, we need the following lemma.Lemma 4.
Letϕ:(0,1]→(0,∞). Suppose that every σ-algebra ℱn is generated by countable atoms. For B∈A(ℱn), let Bj∈A(ℱj) be
(60)B=Bn⊂Bn-1⊂⋯⊂B0.
For the sequence {Bk}k=0n above, one defines a decreasing sequence of integers nj=nj({Bk}k=0n) inductively by
(61)n1=sup{k∈[0,n]∩ℤ:P(Bk)≥2P(B)},nj=sup{k∈[0,nj-1]∩ℤ:P(Bk)≥2P(Bnj-1)}{k∈[0,nj1]∩ℤ:P(Bk)≥2P(Bnj-1)}5(j≥2),
where one uses the convention sup∅=-1. One further defines
(62)J={j:nj≥0},nj+=1+nj.
Suppose that ϕ satisfies the doubling condition. Then, there exists a positive constant C, that depends only on ϕ, such that
(63)∑j∈J‍ϕ(bnj+)≤C∫bn1‍ϕ(t)tdtonB,
where bj is the function defined by (33).Note that this lemma is the counterpart to the technique in [29, page 1104, line 5].Proof.
By the definition ofnj, if j∈J, then
(64)bnj-1≤bnj+<2bnj-1≤bnjonB,
where we use the convention n0=n.
Using the doubling condition onϕ, we have
(65)∑j∈J‍ϕ(bnj+)≲∑j∈J‍∫bnj-12bnj-1‍ϕ(t)tdt≤∫bn1‍ϕ(t)tdt
because the intervals (bnj-1,2bnj-1) are disjointed by (64).In the proof of Theorem19, we need the following estimates for the square function of Iγf.Lemma 5.
Suppose that everyσ-algebra ℱn is generated by countable atoms and that {ℱn}n≥0 is regular. Let p,q∈(0,∞) with p<q. Let (γn)n≥0 be a sequence of nonnegative bounded adapted functions. Suppose that ϕ:(0,1]→(0,∞) satisfies the doubling condition. Assume that there exists a positive constant C such that
(66)∑k=0n‍γk-1ϕ(bk-1)χ{bk≠bk-1}+ϕ(bn)×∑k=n+1∞‍γk-1χ{bk≠bk-1}≤Cϕ(bn)p/qa.s.
for all n≥0, where bk is the measurable function defined by (33). Then, for f∈ℳ with ∥f∥Hp,ϕS=1,
(67)Sn(Iγf)≤Cϕ(bn-1)p/q,S(n)(Iγf)≤Cϕ(bn)p/q-1S(f)
for all n≥0, where C is a positive constant independent of f.Proof.
Letf∈ℳ such that ∥f∥Hp,ϕS=1. We first show that
(68)|dkf|≤Cϕ(bk-1),
where C is a positive constant that depends only on ϕ and the regularity constant R. Let B∈A(ℱk). Then, on the set B, keeping in mind that
(69)|dkf|=(1P(B)∫B‍|dkf|pdP)1/p,
we have
(70)|dkf|≤(1P(B)∫B‍S(f)pdP)1/p≤ϕ(P(B))∥f∥Hp,ϕS≲ϕ(bk-1).
We have obtained (68). We now show (67). Using (68) and the assumption (66), we have
(71)Sn(Iγf)2=∑k=0n‍γk-12|dkf|2≲∑k=0n‍γk-12ϕ(bk-1)2χ{bk≠bk-1}≤(∑k=0n‍γk-1ϕ(bk-1)χ{bk≠bk-1})2≲ϕ(bn)2p/q≲ϕ(bn-1)2p/q,S(n)(Iγf)2=∑k=n+1∞‍γk-12|dkf|2≤S(n)(f)2∑k=n+1∞‍γk-12χ{bk≠bk-1}≤S(n)(f)2(∑k=n+1∞‍γk-1χ{bk≠bk-1})2≲ϕ(bn)2p/q-2S(n)(f)2.Remark 22.
In the course of the proof, the embeddingℓ1↪ℓ2 is used. If one does not use the embedding, then
(72)∑k=0n‍γk-1|dkf|≲ϕ(bn-1)p/q,∑k=n+1∞‍γk-1|dkf|≲ϕ(bn)p/q-1S(n)(f).


## 6. Proofs of the Results in Section3

In this section, we prove the results in Section3.Proposition8 can be proved in the same way as [5, Proposition 2.2], so we omit the proof. Proposition 9 is a direct consequence of Lemma 1. Then, we will prove Theorems 11, 12, and 13.Recall thatS(n)(f) is defined by (11).


### 6.1. Proof of Theorem11

We first show Theorem11, Burkholder’s inequality on generalized martingale Campanato spaces.Proof of Theorem11.
Letf∈L10 and B∈ℬn. Then, fχB-En[fχB]∈L10⊂ℳ and
(73)dk(fχB-En[fχB])={0ifk≤n,(dkf)χBifk>n.
Therefore, we have S(fχB-En[fχB])=S(n)(f)χB. Hence, using Theorem 10, we have
(74)cp(∫B‍|f-Enf|pdP)1/p=cp∥fχB-En[fχB]∥Lp≤∥S(fχB-En[fχB])∥Lp=(∫B‍S(n)(f)pdP)1/p≤Cp∥fχB-En[fχB]∥Lp=Cp(∫B‍|f-Enf|pdP)1/p.
We have obtained (21).
We next show (22). Using (74), we have
(75)(∫B‍|f-En-1f|pdP)1/p≤(∫B‍|f-Enf|pdP)1/p+(∫B‍|dnf|pdP)1/p≤cp-1(∫B‍S(n)(f)pdP)1/p+(∫B‍|dnf|pdP)1/p≤cp-1(∫B‍S(n-1)(f)pdP)1/p+(∫B‍S(n-1)(f)pdP)1/p.
Therefore,
(76)cp1+cp∥f∥ℒp,ϕ-≤∥f∥ℋp,ϕS-.
For the converse part, using the inequality
(77)(∫B‍|f-Enf|pdP)1/p≤2(∫B‍|f-En-1f|pdP)1/p,
which we have mentioned in Remark 4, we obtain
(78)(∫B‍S(n-1)(f)pdP)1/p≤(∫B‍S(n)(f)pdP)1/p+(∫B‍|dnf|pdP)1/p≤Cp(∫B‍|f-Enf|pdP)1/p+(∫B‍|dnf|pdP)1/p≤(2Cp+1)(∫B‍|f-En-1f|pdP)1/p.
That is,
(79)∥f∥ℒp,ϕ-≤(2Cp+1)∥f∥ℋp,ϕS-.


### 6.2. Proof of Theorem12

We next show Theorem12, a relation between ℒp,ϕ and ℒp,ϕ-, ℋp,ϕS, and ℋp,ϕS-.Proof of Theorem12.
Inequality (24) was mentioned in Remark 4. Inequality (25) is deduced from the inequality S(f)2-Sn(f)2≤S(f)2-Sn-1(f)2.
We now show (26). Let B∈ℬn and B′={ω∈Ω:En-1[χB](ω)>0}. Since B′∈ℱn-1, we have
(80)E[χΩ∖B′χB]=E[χΩ∖B′En-1[χB]]=0;
that is, B⊂B′.
Suppose that{ℱn}n≥0 is regular. To show (26), we first prove
(81)B′={ω∈Ω:En-1[χB](ω)≥1R},
where R is the regularity constant. By the definition of B′, we have B′⊃{ω∈Ω:En-1[χB](ω)≥1/R}. We will show the converse. By the regularity, we have χB≤REn-1[χB]. This implies B⊂{ω∈Ω:En-1[χB](ω)≥1/R}, or equivalently,
(82)χB≤χ{En-1[χB]≥1/R}.
Operating En-1, we have
(83)En-1[χB]≤χ{En-1[χB]≥1/R}.
We have obtained (81).
From (81), we deduce that
(84)P(B′)=E[χ{En-1[χB]≥1/R}]≤E[REn-1[χB]]=RP(B).
Hence, using the assumption (23) and the doubling condition on ϕ with (84), we have
(85)1P(B)∫B‍|f-En-1f|pdP≤1P(B)∫B′‍|f-En-1f|pdP≲1P(B′)∫B′‍|f-En-1f|pdP≤ϕ(P(B′))p∥f∥ℒp,ϕp≲ϕ(P(B))p∥f∥ℒp,ϕp.
We have obtained (26).
By the same way as above, we have (27). The proof is completed.


### 6.3. Proof of Theorem13

We now prove Theorem13, a relation between martingale Morrey spaces and martingale Campanato spaces.Proof of Theorem13.
The part∥f∥ℒp,ϕ≤2∥f∥Lp,ϕ was shown in Remark 4, and the part ∥f∥ℋp,ϕS-≤∥f∥Hp,ϕS is obvious. We now show the part ∥f∥Lp,ϕ≲∥f∥ℒp,ϕ-.
LetB∈A(ℱn). We take Bk∈A(ℱk) such that B=Bn⊂Bn-1⊂⋯⊂B0. Let nj be the decreasing sequence of integers defined in Lemma 4, with convention n0=n. Since nj-1≥nj, the function Enj-1f-Enjf is constant on Bnj-1. Therefore, on the set B, we have
(86)|Enj-1f-Enjf|=(1P(Bnj-1)∫Bnj-1‍|Enj-1f-Enjf|pdP)1/p≤21/p(1P(Bnj+)∫Bnj+‍|Enj-1f-Enjf|pdP)1/p≤21/p∥f∥ℒp,ϕ-ϕ(P(Bnj+)),
where nj+ is the same as in  (62).
LetJ be the same as in Lemma 4 and let m=maxJ. Using Lemma 4 and the assumption (28), we have
(87)|Enf-Enmf|≤∑j∈J‍|Enj-1f-Enjf|≲∥f∥ℒp,ϕ-∑j∈J‍ϕ(P(Bnj+))≲∥f∥ℒp,ϕ-∫P(B)1‍ϕ(t)tdt≲∥f∥ℒp,ϕ-ϕ(P(B))onB.
For |Enmf|, we may assume that nm>0. By the definition of nm, we have P(B1)≤P(B0)<2P(Bnm)≤2P(B1)≤2P(B0). Therefore,
(88)ϕ(P(B1))≲∫P(B0)/2P(B0)‍ϕ(t)tdt≲∫P(B)1‍ϕ(t)tdt≲ϕ(P(B)).
Hence, on the set B, the constant Enmf has the following bound:
(89)|Enmf|=|Enmf-E0f|=(1P(Bnm)∫Bnm‍|Enmf-E0f|pdP)1/p≤21/p(1P(B1)∫B1‍|Enmf-E0f|pdP)1/p≤21/p∥f∥ℒp,ϕ-ϕ(P(B1))≲∥f∥ℒp,ϕ-ϕ(P(B)).
Combining (87) and (89), we have
(90)|Enf|≲∥f∥ℒp,ϕ-ϕ(P(B))onB.
Using (24) in Theorem 12, we have
(91)(∫B‍|f|pdP)1/p≤(∫B‍|f-Enf|pdP)1/p+P(B)1/p|Enf|≲P(B)1/pϕ(P(B))∥f∥ℒp,ϕ-+P(B)1/pϕ(P(B))∥f∥ℒp,ϕ-~P(B)1/pϕ(P(B))∥f∥ℒp,ϕ-;
that is,
(92)∥f∥Lp,ϕ≲∥f∥ℒp,ϕ-.
We can show ∥f∥Hp,ϕS≲∥f∥ℋp,ϕS- by the same way. Indeed, in (86), we can replace |Enj-1f-Enjf| and ∥f∥ℒp,ϕ- by {Snj-1(f)2-Snj(f)2}1/2 and ∥f∥ℋp,ϕS-, respectively. The rest is similar and we can obtain ∥f∥Hp,ϕS≲∥f∥ℋp,ϕS-.
Ifℬ=𝒜, then ℬ satisfies (23). Therefore, if ℱ is regular, we can apply Theorem 12 to obtain the equivalence of ∥f∥Lp,ϕ, ∥f∥ℒp,ϕ, and ∥f∥ℒp,ϕ- and the equivalence of ∥f∥Hp,ϕS, ∥f∥ℋp,ϕS, and ∥f∥ℋp,ϕS-.


### 6.4. Proof of Theorem15

We will now prove Theorem15, the John-Nirenberg-type theorem for martingale Campanato-Hardy spaces. Following Weisz [19, Definition 2.45], we define
(93)∥f∥BMOpS=supn≥0∥(En[{S(f)2-Sn-1(f)2}p/2])1/p∥L∞,
for f∈ℳ and p∈(0,∞).Proof of Theorem15.
We may assume thatℬ=ℱ by Proposition 9.
By Hölder’s inequality and Theorem12, we only need to show that ∥f∥ℋp,ϕS≤Cp,q,ϕ∥f∥ℋq,ϕS- for 0<q≤1<p.
Recall the notationS(n)(f)2=S(f)2-Sn(f)2. Let f∈ℋq,ϕS-∩L10⊂ℳ, A∈ℱn, and m≥n+1. By (73), we have S(m-1)(fχA-En[fχA])=S(m-1)(f)χA. Hence, for B∈ℱm, m≥n+1, we have
(94)1P(B)∫B‍S(m-1)(fχA-En[fχA])qdP=1P(B)∫A∩B‍S(m-1)(f)qdP≤1P(A∩B)∫A∩B‍S(m-1)(f)qdP≤ϕ(P(A∩B))q∥f∥ℋq,ϕS-q≲ϕ(P(A))q∥f∥ℋq,ϕS-q.
Therefore, for the {ℱm}m≥n-martingale (Em[fχA]-En[fχA])m≥n, we have
(95)∥(Em[fχA]-En[fχA])m≥n∥BMOqS≲ϕ(P(A))∥f∥ℋq,ϕS-.
By [19, Theorem 2.50], there exists a positive constant Cp,q,ϕ that depends only on p, q, and ϕ such that
(96)(En+1[S(n)(fχA-En[fχA])p])1/p≤Cp,q,ϕϕ(P(A))∥f∥ℋq,ϕS-.
Combining (96) and the fact that S(n)(fχA-En[fχA])=S(n)(f)χA, we have
(97)(En+1[S(n)(f)p])1/pχA≤Cp,q,ϕϕ(P(A))∥f∥ℋq,ϕS-.
Therefore, for A∈ℱn, we have
(98)(1P(A)∫A‍S(n)(f)pdP)1/p=(1P(A)∫A‍En+1[S(n)(f)p]dP)1/p≤Cp,q,ϕϕ(P(A))∥f∥ℋq,ϕS-;
that is,
(99)∥f∥ℋp,ϕS≤Cp,q,ϕ∥f∥ℋq,ϕS-
for f∈ℋq,ϕS-∩L10. For general f∈ℋq,ϕS-, applying (99) to the martingale f(m)=(fmin(m,n))n≥0, we have
(100)∥f(m)∥ℋp,ϕS≤Cp,q,ϕ∥f(m)∥ℋq,ϕS-≤Cp,q,ϕ∥f∥ℋq,ϕS-.
Taking p=2 in (100), we have that f is an L2-bounded martingale. Therefore, we have (99) for all f∈ℋq,ϕS-. The proof is completed.


## 6.1. Proof of Theorem11

We first show Theorem11, Burkholder’s inequality on generalized martingale Campanato spaces.Proof of Theorem11.
Letf∈L10 and B∈ℬn. Then, fχB-En[fχB]∈L10⊂ℳ and
(73)dk(fχB-En[fχB])={0ifk≤n,(dkf)χBifk>n.
Therefore, we have S(fχB-En[fχB])=S(n)(f)χB. Hence, using Theorem 10, we have
(74)cp(∫B‍|f-Enf|pdP)1/p=cp∥fχB-En[fχB]∥Lp≤∥S(fχB-En[fχB])∥Lp=(∫B‍S(n)(f)pdP)1/p≤Cp∥fχB-En[fχB]∥Lp=Cp(∫B‍|f-Enf|pdP)1/p.
We have obtained (21).
We next show (22). Using (74), we have
(75)(∫B‍|f-En-1f|pdP)1/p≤(∫B‍|f-Enf|pdP)1/p+(∫B‍|dnf|pdP)1/p≤cp-1(∫B‍S(n)(f)pdP)1/p+(∫B‍|dnf|pdP)1/p≤cp-1(∫B‍S(n-1)(f)pdP)1/p+(∫B‍S(n-1)(f)pdP)1/p.
Therefore,
(76)cp1+cp∥f∥ℒp,ϕ-≤∥f∥ℋp,ϕS-.
For the converse part, using the inequality
(77)(∫B‍|f-Enf|pdP)1/p≤2(∫B‍|f-En-1f|pdP)1/p,
which we have mentioned in Remark 4, we obtain
(78)(∫B‍S(n-1)(f)pdP)1/p≤(∫B‍S(n)(f)pdP)1/p+(∫B‍|dnf|pdP)1/p≤Cp(∫B‍|f-Enf|pdP)1/p+(∫B‍|dnf|pdP)1/p≤(2Cp+1)(∫B‍|f-En-1f|pdP)1/p.
That is,
(79)∥f∥ℒp,ϕ-≤(2Cp+1)∥f∥ℋp,ϕS-.


## 6.2. Proof of Theorem12

We next show Theorem12, a relation between ℒp,ϕ and ℒp,ϕ-, ℋp,ϕS, and ℋp,ϕS-.Proof of Theorem12.
Inequality (24) was mentioned in Remark 4. Inequality (25) is deduced from the inequality S(f)2-Sn(f)2≤S(f)2-Sn-1(f)2.
We now show (26). Let B∈ℬn and B′={ω∈Ω:En-1[χB](ω)>0}. Since B′∈ℱn-1, we have
(80)E[χΩ∖B′χB]=E[χΩ∖B′En-1[χB]]=0;
that is, B⊂B′.
Suppose that{ℱn}n≥0 is regular. To show (26), we first prove
(81)B′={ω∈Ω:En-1[χB](ω)≥1R},
where R is the regularity constant. By the definition of B′, we have B′⊃{ω∈Ω:En-1[χB](ω)≥1/R}. We will show the converse. By the regularity, we have χB≤REn-1[χB]. This implies B⊂{ω∈Ω:En-1[χB](ω)≥1/R}, or equivalently,
(82)χB≤χ{En-1[χB]≥1/R}.
Operating En-1, we have
(83)En-1[χB]≤χ{En-1[χB]≥1/R}.
We have obtained (81).
From (81), we deduce that
(84)P(B′)=E[χ{En-1[χB]≥1/R}]≤E[REn-1[χB]]=RP(B).
Hence, using the assumption (23) and the doubling condition on ϕ with (84), we have
(85)1P(B)∫B‍|f-En-1f|pdP≤1P(B)∫B′‍|f-En-1f|pdP≲1P(B′)∫B′‍|f-En-1f|pdP≤ϕ(P(B′))p∥f∥ℒp,ϕp≲ϕ(P(B))p∥f∥ℒp,ϕp.
We have obtained (26).
By the same way as above, we have (27). The proof is completed.


## 6.3. Proof of Theorem13

We now prove Theorem13, a relation between martingale Morrey spaces and martingale Campanato spaces.Proof of Theorem13.
The part∥f∥ℒp,ϕ≤2∥f∥Lp,ϕ was shown in Remark 4, and the part ∥f∥ℋp,ϕS-≤∥f∥Hp,ϕS is obvious. We now show the part ∥f∥Lp,ϕ≲∥f∥ℒp,ϕ-.
LetB∈A(ℱn). We take Bk∈A(ℱk) such that B=Bn⊂Bn-1⊂⋯⊂B0. Let nj be the decreasing sequence of integers defined in Lemma 4, with convention n0=n. Since nj-1≥nj, the function Enj-1f-Enjf is constant on Bnj-1. Therefore, on the set B, we have
(86)|Enj-1f-Enjf|=(1P(Bnj-1)∫Bnj-1‍|Enj-1f-Enjf|pdP)1/p≤21/p(1P(Bnj+)∫Bnj+‍|Enj-1f-Enjf|pdP)1/p≤21/p∥f∥ℒp,ϕ-ϕ(P(Bnj+)),
where nj+ is the same as in  (62).
LetJ be the same as in Lemma 4 and let m=maxJ. Using Lemma 4 and the assumption (28), we have
(87)|Enf-Enmf|≤∑j∈J‍|Enj-1f-Enjf|≲∥f∥ℒp,ϕ-∑j∈J‍ϕ(P(Bnj+))≲∥f∥ℒp,ϕ-∫P(B)1‍ϕ(t)tdt≲∥f∥ℒp,ϕ-ϕ(P(B))onB.
For |Enmf|, we may assume that nm>0. By the definition of nm, we have P(B1)≤P(B0)<2P(Bnm)≤2P(B1)≤2P(B0). Therefore,
(88)ϕ(P(B1))≲∫P(B0)/2P(B0)‍ϕ(t)tdt≲∫P(B)1‍ϕ(t)tdt≲ϕ(P(B)).
Hence, on the set B, the constant Enmf has the following bound:
(89)|Enmf|=|Enmf-E0f|=(1P(Bnm)∫Bnm‍|Enmf-E0f|pdP)1/p≤21/p(1P(B1)∫B1‍|Enmf-E0f|pdP)1/p≤21/p∥f∥ℒp,ϕ-ϕ(P(B1))≲∥f∥ℒp,ϕ-ϕ(P(B)).
Combining (87) and (89), we have
(90)|Enf|≲∥f∥ℒp,ϕ-ϕ(P(B))onB.
Using (24) in Theorem 12, we have
(91)(∫B‍|f|pdP)1/p≤(∫B‍|f-Enf|pdP)1/p+P(B)1/p|Enf|≲P(B)1/pϕ(P(B))∥f∥ℒp,ϕ-+P(B)1/pϕ(P(B))∥f∥ℒp,ϕ-~P(B)1/pϕ(P(B))∥f∥ℒp,ϕ-;
that is,
(92)∥f∥Lp,ϕ≲∥f∥ℒp,ϕ-.
We can show ∥f∥Hp,ϕS≲∥f∥ℋp,ϕS- by the same way. Indeed, in (86), we can replace |Enj-1f-Enjf| and ∥f∥ℒp,ϕ- by {Snj-1(f)2-Snj(f)2}1/2 and ∥f∥ℋp,ϕS-, respectively. The rest is similar and we can obtain ∥f∥Hp,ϕS≲∥f∥ℋp,ϕS-.
Ifℬ=𝒜, then ℬ satisfies (23). Therefore, if ℱ is regular, we can apply Theorem 12 to obtain the equivalence of ∥f∥Lp,ϕ, ∥f∥ℒp,ϕ, and ∥f∥ℒp,ϕ- and the equivalence of ∥f∥Hp,ϕS, ∥f∥ℋp,ϕS, and ∥f∥ℋp,ϕS-.


## 6.4. Proof of Theorem15

We will now prove Theorem15, the John-Nirenberg-type theorem for martingale Campanato-Hardy spaces. Following Weisz [19, Definition 2.45], we define
(93)∥f∥BMOpS=supn≥0∥(En[{S(f)2-Sn-1(f)2}p/2])1/p∥L∞,
for f∈ℳ and p∈(0,∞).Proof of Theorem15.
We may assume thatℬ=ℱ by Proposition 9.
By Hölder’s inequality and Theorem12, we only need to show that ∥f∥ℋp,ϕS≤Cp,q,ϕ∥f∥ℋq,ϕS- for 0<q≤1<p.
Recall the notationS(n)(f)2=S(f)2-Sn(f)2. Let f∈ℋq,ϕS-∩L10⊂ℳ, A∈ℱn, and m≥n+1. By (73), we have S(m-1)(fχA-En[fχA])=S(m-1)(f)χA. Hence, for B∈ℱm, m≥n+1, we have
(94)1P(B)∫B‍S(m-1)(fχA-En[fχA])qdP=1P(B)∫A∩B‍S(m-1)(f)qdP≤1P(A∩B)∫A∩B‍S(m-1)(f)qdP≤ϕ(P(A∩B))q∥f∥ℋq,ϕS-q≲ϕ(P(A))q∥f∥ℋq,ϕS-q.
Therefore, for the {ℱm}m≥n-martingale (Em[fχA]-En[fχA])m≥n, we have
(95)∥(Em[fχA]-En[fχA])m≥n∥BMOqS≲ϕ(P(A))∥f∥ℋq,ϕS-.
By [19, Theorem 2.50], there exists a positive constant Cp,q,ϕ that depends only on p, q, and ϕ such that
(96)(En+1[S(n)(fχA-En[fχA])p])1/p≤Cp,q,ϕϕ(P(A))∥f∥ℋq,ϕS-.
Combining (96) and the fact that S(n)(fχA-En[fχA])=S(n)(f)χA, we have
(97)(En+1[S(n)(f)p])1/pχA≤Cp,q,ϕϕ(P(A))∥f∥ℋq,ϕS-.
Therefore, for A∈ℱn, we have
(98)(1P(A)∫A‍S(n)(f)pdP)1/p=(1P(A)∫A‍En+1[S(n)(f)p]dP)1/p≤Cp,q,ϕϕ(P(A))∥f∥ℋq,ϕS-;
that is,
(99)∥f∥ℋp,ϕS≤Cp,q,ϕ∥f∥ℋq,ϕS-
for f∈ℋq,ϕS-∩L10. For general f∈ℋq,ϕS-, applying (99) to the martingale f(m)=(fmin(m,n))n≥0, we have
(100)∥f(m)∥ℋp,ϕS≤Cp,q,ϕ∥f(m)∥ℋq,ϕS-≤Cp,q,ϕ∥f∥ℋq,ϕS-.
Taking p=2 in (100), we have that f is an L2-bounded martingale. Therefore, we have (99) for all f∈ℋq,ϕS-. The proof is completed.


## 7. Proofs of the Results in Section4

In this section, we prove the results in Section4.


### 7.1. Proofs of Theorem16 and Corollary 17

Recall thatS(n)(f)2=S(f)2-Sn(f)2.Proof of Theorem16.
Using the assumption that(γn)n≥0 is almost decreasing, we have
(101)S(n)(Iγf)2=∑k=n+1∞‍|γk-1dkf|2≤Aγ2γn2∑k=n+1∞‍|dkf|2=Aγ2γn2S(n)(f)2.
Then, for B∈ℬn, using the assumption (39), we have
(102)∫B‍S(n)(Iγf)pdP≤Aγp∫B‍γnpS(n)(f)pdP≤Aγp∥γnχB∥L∞p∫B‍S(n)(f)pdP≤Aγp∥γnχB∥L∞pP(B)ϕ(P(B))p∥f∥ℋp,ϕSp≤AγpCγ,ϕ,ψpP(B)ψ(P(B))p∥f∥ℋp,ϕSp.
Therefore, we have
(103)∥Iγf∥ℋp,ψS≤AγCγ,ϕ,ψ∥f∥ℋp,ϕS
and Iγ∈B(ℋp,ϕS,ℋp,ψS). The proof is completed.Proof of Corollary17.
Letγn=ρ(bn). Then, we have that (γn)n≥0 is almost decreasing and that ∥γnχB∥L∞=ρ(P(B)) for B∈A(ℱn). Hence,
(104)Cγ,ϕ,ψ=supn≥0supB∈A(ℱn)ρ(P(B))ϕ(P(B))ψ(P(B))≤sup0<t≤1ρ(t)ϕ(t)ψ(t)<∞.
Therefore, we can apply Theorem 16 to obtain Iγ∈B(ℋp,ϕS,ℋp,ψS). If we further assume that {ℱn}n≥0 is regular and that ψ is almost increasing and satisfies the doubling condition, then, by the John-Nirenberg-type equivalence (Theorem 15), we have Iγ∈B(ℋp,ϕS,ℋq,ψS) for all q∈(0,∞).Remark 23.
Assume that (39) holds. Suppose further that there exists a positive number C′ such that ∑k=n+1∞‍γk-1≤C′γn a.s. for all n≥0. Then, in the light of Remark 22, we see that
(105)supn≥0supB∈ℬn1ψ(P(B))(×∫B‍(∑k=n+1∞‍|γk-1dkf|)p1P(B)∫B‍(∑k=n+1∞‍|γk-1dkf|)pdP)1/p≤C′Cγ,ϕ,ψ∥f∥ℋp,ϕS.


### 7.2. Proofs of Theorem19 and Corollaries 20 and 21

Proof of Theorem19.
We first show the partIγ∈B(Hp,ϕS,Hq,ϕp/qS). Assume (45). Let f=(fn)n≥0∈ℳ such that ∥f∥Hp,ϕS=1. We need only to show that there exists C>0 independent of f such that
(106)∥Iγf∥Hq,ϕp/qS≤C.
To obtain (106), we first show that
(107)S(Iγf)≤CS(f)p/q.
Let N=∑n=0∞‍χ{ϕ(bn)≤S(f)}. We define measurable subsets Ω1, Ω2, and Ω3 by
(108)Ω1={N=∞},Ω2={N=0},Ω3={0<N<∞}.
Let ω∈Ω1. Then, we can take infinitely many integers n such that ϕ(bn-1(ω))≤S(f)(ω). For such n, we have
(109)Sn(Iγf)(ω)≤Cϕ(bn-1(ω))p/q≤CS(f)(ω)p/q
by Lemma 5. Letting n→∞ along n that satisfies ϕ(bn-1(ω))≤S(f)(ω), we have (107) on Ω1.
OnΩ2, again by Lemma 5, we have
(110)S(Iγf)≤Cϕ(b0)p/q-1S(f)≤CS(f)p/q-1S(f)=CS(f)p/q.
Let ω∈Ω3. Then, we can take an integer n such that
(111)ϕ(bn-1(ω))≤S(f)(ω),ϕ(bn(ω))>S(f)(ω).
Hence, by Lemma 5, we have
(112)S(Iγf)(ω)≤Sn(Iγf)(ω)+S(n)(Iγf)(ω)≲ϕ(bn-1(ω))p/q+ϕ(bn(ω))p/q-1S(f)(ω)≤S(f)(ω)p/q+S(f)(ω)p/q-1S(f)(ω)≲S(f)(ω)p/q.
We have obtained (107).
We now show (106). Let B∈∪nA(ℱn). Using (107), we have
(113)(∫B‍S(Iγf)qdP)1/q≲{(∫B‍S(f)pdP)1/p}p/q≤P(B)1/qϕ(P(B))p/q∥f∥Hp,ϕSp/q=P(B)1/qϕ(P(B))p/q.
We have obtained (106).
We now show the partIγ∈B(HpS,HqS). Let ϕ(t)=t-1/p. We simply denote Hp,ϕS by Hp,-1/pS. Let f=(fn)n≥0∈ℳ such that ∥f∥HpS=1. Observe that ∥f∥Hp,-1/pS≤∥f∥HpS. By the assumption that (45) holds for ϕ(t)=t-1/p, we can apply (107) to f/∥f∥Hp,-1/pS, and we have
(114)S(Iγf)≤CS(f)p/q∥f∥Hp,-1/pS1-p/q≤CS(f)p/q.
Hence, we obtain
(115)(∫Ω‍S(Iγf)qdP)1/q≲{(∫Ω‍S(f)pdP)1/p}p/q=1.
The proof is completed.Proof of Corollary20.
Letγn=ρ(bn). We only have to verify (45). Using Lemma 3 and the assumption (48), we have
(116)∑k=0n‍ρ(bk-1)ϕ(bk-1)χ{bk≠bk-1}+ϕ(bn)∑k=n+1∞‍ρ(bk-1)χ{bk≠bk-1}≲∫bn1‍ρ(t)ϕ(t)tdt+ϕ(bn)∫0bn‍ρ(t)tdt≲ϕ(bn)p/q.
By Theorem 19, we have the conclusion.Proof of Corollary21.
Ifρ(r)=rα and ϕ(t)=t-1/p, then
(117)∫r1‍ρ(t)ϕ(t)tdt+ϕ(r)∫0r‍ρ(t)tdt~rα-1/p=r-1/q=ϕ(r)p/q.
Observing (116) and applying Theorem 19 to Iα, we have the conclusion.Remark 24.
In the light of Remark22, we see that
(118)supn≥0supB∈A(ℱn)1ϕ(P(B))p/q×(1P(B)∫B‍(∑k=1∞‍|γk-1dkf|)qdP)1/q≤C∥f∥Hp,ϕS,
which is similar to (105).
In words of harmonic analysis, this corresponds to the embeddingFp1q1s+n/p1-n/p2↪Fp2q2s for 0<p1<p2<∞, 0<q1,q2≤∞, and s∈ℝ; see [30, Section 2.3] and [30, page 129] for the definition of the space Fpqs and the above embedding, respectively. It may be interesting to observe that this embedding is translated into the fact that Iα makes functions have bounded variation.


### 7.3.  Proof of Proposition18

In this subsection, we prove Proposition18.Proof of Proposition18.
To proveIγ∉B(Hp,ϕS,Hq,ϕp/qS)∖{0}, we only need to show the following for any f=(fn)n≥0∈ℳ:
(119)ifIγ∈B(Hp,ϕS,Hq,ϕp/qS),thenχ{|γk-1|>0}dkf=0ifIγ∈B(Hp,ϕS,Hq,ϕp/qS),then5555(k=1,2,…).
We now show (119) for f=(fn)n≥0∈ℳ. We may assume that P(|γk-1|>0)>0. For γk-1, define
(120)ℱk+({|γk-1|>0})={B∈ℱk∖ℱk-1:P({|γk-1|>0}∩B)>0}.
If ℱk+({|γk-1|>0})=∅, then the function χ{|γk-1|>0}dkf is ℱk-1-measurable. Therefore, we have
(121)χ{|γk-1|>0}dkf=Ek-1[χ{|γk-1|>0}dkf]=χ{|γk-1|>0}Ek-1[dkf]=0.
To complete the proof of (119), we only have to show the following:
(122)ifℱk+({|γk-1|>0})≠∅,thenIγ∉B(Hp,ϕS,Hq,ϕp/qS).
Assume that ℱk+({|γk-1|>0})≠∅. We fix B∈ℱk∖ℱk-1 and δ>0 such that P({|γk-1|≥δ}∩B)>0. Let B1={|γk-1|≥δ} and B1′={|γk-1|≥δ}∩B. Note that B1∈ℱk-1, B1′∈ℱk+({|γk-1|>0}), and B1′⊂B1.
To prove (122), we define two decreasing sequences of measurable sets {Bn}n=1∞ and {Bn′}n=1∞ that satisfy
(123)Bn∈ℱk-1,P(Bn)=P(B1)2n-1,Bn′∈ℱk+({|γk-1|>0}),Bn′⊂Bn
for every n≥1, inductively as follows.
Suppose that we can chooseBn-1 and Bn-1′ that satisfy
(124)Bn-1∈ℱk-1,P(Bn-1)=P(B1)2n-2,Bn-1′∈ℱk+({|γk-1|>0}),Bn-1′⊂Bn-1.
By the assumption that ℱ0 is nonatomic, ℱk-1 is also nonatomic. Hence, there exists Bn∈ℱk-1 such that Bn⊂Bn-1, P(Bn)=P(Bn-1)/2 and P(Bn∩Bn-1′)>0. Let Bn′=Bn∩Bn-1′. Then, we have Bn and Bn′ with Bn⊂Bn-1 and Bn′⊂Bn-1′ that satisfy (123).
For the setBn′ defined above, let gn=χBn′-Ek-1[χBn′]. Since Bn is ℱk-1-measurable, we have
(125)gnχΩ∖Bn=(χBn′-Ek-1[χBn′])χΩ∖Bn=χBn′χΩ∖Bn-Ek-1[χBn′χΩ∖Bn]=0.
By (125) and the assumption that t1/pϕ(t) is almost increasing, we have, for any A∈∪n=0∞ℱn,
(126)1ϕ(P(A))(1P(A)∫A‍|gn|pdP)1/p=1ϕ(P(A))P(A)1/p(∫A∩Bn‍|gn|pdP)1/p≲1ϕ(P(A∩Bn))P(A∩Bn)1/p×(∫A∩Bn‍|gn|pdP)1/p=1ϕ(P(A∩Bn))×(1P(A∩Bn)∫A∩Bn‍|gn|pdP)1/p.
We now show that Iγ∉B(Hp,ϕS,Hq,ϕp/qS). Suppose that Iγ∈B(Hp,ϕS,Hq,ϕp/qS); that is, there exists a positive number C such that
(127)∥Iγf∥Hq,ϕp/qS≤C∥f∥Hp,ϕS
for all f∈Hp,ϕS.
SinceBn′∈ℱk∖ℱk-1, we have dkgn=gn≠0, and djgn=0 for j≠k.
Therefore, we have(128)S(gn)=gn,S(Iγgn)=γk-1gn.
Forgn, we take Dn∈∪n=0∞ℱn such that
(129)1ϕ(P(Dn))(1P(Dn)∫Dn‍|gn|pdP)1/p≥12∥gn∥Hp,ϕS.
By (126), we may assume that Dn⊂Bn. As a consequence, we have Dn⊂B1={|γk-1|≥δ}, and limn→∞P(Dn)=0 by (123). Then, using (127) with (128), we have
(130)1ϕ(P(Dn))p/q(1P(Dn)∫Dn‍|gn|qdP)1/q≤1δ1ϕ(P(Dn))p/q×(1P(Dn)∫Dn‍|γk-1gn|qdP)1/q≤1δ∥Iγgn∥Hq,ϕp/qS≤Cδ∥gn∥Hp,ϕS≤2Cδ1ϕ(P(Dn))×(1P(Dn)∫Dn‍|gn|pdP)1/p≤2Cδ1ϕ(P(Dn))×(1P(Dn)∫Dn‍|gn|qdP)1/q.
Therefore, we have
(131)ϕ(P(Dn))1-p/q≤2Cδ.
However, this contradicts p<q and limt→0ϕ(t)=∞. We have (122) and hence have (119).


## 7.1. Proofs of Theorem16 and Corollary 17

Recall thatS(n)(f)2=S(f)2-Sn(f)2.Proof of Theorem16.
Using the assumption that(γn)n≥0 is almost decreasing, we have
(101)S(n)(Iγf)2=∑k=n+1∞‍|γk-1dkf|2≤Aγ2γn2∑k=n+1∞‍|dkf|2=Aγ2γn2S(n)(f)2.
Then, for B∈ℬn, using the assumption (39), we have
(102)∫B‍S(n)(Iγf)pdP≤Aγp∫B‍γnpS(n)(f)pdP≤Aγp∥γnχB∥L∞p∫B‍S(n)(f)pdP≤Aγp∥γnχB∥L∞pP(B)ϕ(P(B))p∥f∥ℋp,ϕSp≤AγpCγ,ϕ,ψpP(B)ψ(P(B))p∥f∥ℋp,ϕSp.
Therefore, we have
(103)∥Iγf∥ℋp,ψS≤AγCγ,ϕ,ψ∥f∥ℋp,ϕS
and Iγ∈B(ℋp,ϕS,ℋp,ψS). The proof is completed.Proof of Corollary17.
Letγn=ρ(bn). Then, we have that (γn)n≥0 is almost decreasing and that ∥γnχB∥L∞=ρ(P(B)) for B∈A(ℱn). Hence,
(104)Cγ,ϕ,ψ=supn≥0supB∈A(ℱn)ρ(P(B))ϕ(P(B))ψ(P(B))≤sup0<t≤1ρ(t)ϕ(t)ψ(t)<∞.
Therefore, we can apply Theorem 16 to obtain Iγ∈B(ℋp,ϕS,ℋp,ψS). If we further assume that {ℱn}n≥0 is regular and that ψ is almost increasing and satisfies the doubling condition, then, by the John-Nirenberg-type equivalence (Theorem 15), we have Iγ∈B(ℋp,ϕS,ℋq,ψS) for all q∈(0,∞).Remark 23.
Assume that (39) holds. Suppose further that there exists a positive number C′ such that ∑k=n+1∞‍γk-1≤C′γn a.s. for all n≥0. Then, in the light of Remark 22, we see that
(105)supn≥0supB∈ℬn1ψ(P(B))(×∫B‍(∑k=n+1∞‍|γk-1dkf|)p1P(B)∫B‍(∑k=n+1∞‍|γk-1dkf|)pdP)1/p≤C′Cγ,ϕ,ψ∥f∥ℋp,ϕS.


## 7.2. Proofs of Theorem19 and Corollaries 20 and 21

Proof of Theorem19.
We first show the partIγ∈B(Hp,ϕS,Hq,ϕp/qS). Assume (45). Let f=(fn)n≥0∈ℳ such that ∥f∥Hp,ϕS=1. We need only to show that there exists C>0 independent of f such that
(106)∥Iγf∥Hq,ϕp/qS≤C.
To obtain (106), we first show that
(107)S(Iγf)≤CS(f)p/q.
Let N=∑n=0∞‍χ{ϕ(bn)≤S(f)}. We define measurable subsets Ω1, Ω2, and Ω3 by
(108)Ω1={N=∞},Ω2={N=0},Ω3={0<N<∞}.
Let ω∈Ω1. Then, we can take infinitely many integers n such that ϕ(bn-1(ω))≤S(f)(ω). For such n, we have
(109)Sn(Iγf)(ω)≤Cϕ(bn-1(ω))p/q≤CS(f)(ω)p/q
by Lemma 5. Letting n→∞ along n that satisfies ϕ(bn-1(ω))≤S(f)(ω), we have (107) on Ω1.
OnΩ2, again by Lemma 5, we have
(110)S(Iγf)≤Cϕ(b0)p/q-1S(f)≤CS(f)p/q-1S(f)=CS(f)p/q.
Let ω∈Ω3. Then, we can take an integer n such that
(111)ϕ(bn-1(ω))≤S(f)(ω),ϕ(bn(ω))>S(f)(ω).
Hence, by Lemma 5, we have
(112)S(Iγf)(ω)≤Sn(Iγf)(ω)+S(n)(Iγf)(ω)≲ϕ(bn-1(ω))p/q+ϕ(bn(ω))p/q-1S(f)(ω)≤S(f)(ω)p/q+S(f)(ω)p/q-1S(f)(ω)≲S(f)(ω)p/q.
We have obtained (107).
We now show (106). Let B∈∪nA(ℱn). Using (107), we have
(113)(∫B‍S(Iγf)qdP)1/q≲{(∫B‍S(f)pdP)1/p}p/q≤P(B)1/qϕ(P(B))p/q∥f∥Hp,ϕSp/q=P(B)1/qϕ(P(B))p/q.
We have obtained (106).
We now show the partIγ∈B(HpS,HqS). Let ϕ(t)=t-1/p. We simply denote Hp,ϕS by Hp,-1/pS. Let f=(fn)n≥0∈ℳ such that ∥f∥HpS=1. Observe that ∥f∥Hp,-1/pS≤∥f∥HpS. By the assumption that (45) holds for ϕ(t)=t-1/p, we can apply (107) to f/∥f∥Hp,-1/pS, and we have
(114)S(Iγf)≤CS(f)p/q∥f∥Hp,-1/pS1-p/q≤CS(f)p/q.
Hence, we obtain
(115)(∫Ω‍S(Iγf)qdP)1/q≲{(∫Ω‍S(f)pdP)1/p}p/q=1.
The proof is completed.Proof of Corollary20.
Letγn=ρ(bn). We only have to verify (45). Using Lemma 3 and the assumption (48), we have
(116)∑k=0n‍ρ(bk-1)ϕ(bk-1)χ{bk≠bk-1}+ϕ(bn)∑k=n+1∞‍ρ(bk-1)χ{bk≠bk-1}≲∫bn1‍ρ(t)ϕ(t)tdt+ϕ(bn)∫0bn‍ρ(t)tdt≲ϕ(bn)p/q.
By Theorem 19, we have the conclusion.Proof of Corollary21.
Ifρ(r)=rα and ϕ(t)=t-1/p, then
(117)∫r1‍ρ(t)ϕ(t)tdt+ϕ(r)∫0r‍ρ(t)tdt~rα-1/p=r-1/q=ϕ(r)p/q.
Observing (116) and applying Theorem 19 to Iα, we have the conclusion.Remark 24.
In the light of Remark22, we see that
(118)supn≥0supB∈A(ℱn)1ϕ(P(B))p/q×(1P(B)∫B‍(∑k=1∞‍|γk-1dkf|)qdP)1/q≤C∥f∥Hp,ϕS,
which is similar to (105).
In words of harmonic analysis, this corresponds to the embeddingFp1q1s+n/p1-n/p2↪Fp2q2s for 0<p1<p2<∞, 0<q1,q2≤∞, and s∈ℝ; see [30, Section 2.3] and [30, page 129] for the definition of the space Fpqs and the above embedding, respectively. It may be interesting to observe that this embedding is translated into the fact that Iα makes functions have bounded variation.


## 7.3.  Proof of Proposition18

In this subsection, we prove Proposition18.Proof of Proposition18.
To proveIγ∉B(Hp,ϕS,Hq,ϕp/qS)∖{0}, we only need to show the following for any f=(fn)n≥0∈ℳ:
(119)ifIγ∈B(Hp,ϕS,Hq,ϕp/qS),thenχ{|γk-1|>0}dkf=0ifIγ∈B(Hp,ϕS,Hq,ϕp/qS),then5555(k=1,2,…).
We now show (119) for f=(fn)n≥0∈ℳ. We may assume that P(|γk-1|>0)>0. For γk-1, define
(120)ℱk+({|γk-1|>0})={B∈ℱk∖ℱk-1:P({|γk-1|>0}∩B)>0}.
If ℱk+({|γk-1|>0})=∅, then the function χ{|γk-1|>0}dkf is ℱk-1-measurable. Therefore, we have
(121)χ{|γk-1|>0}dkf=Ek-1[χ{|γk-1|>0}dkf]=χ{|γk-1|>0}Ek-1[dkf]=0.
To complete the proof of (119), we only have to show the following:
(122)ifℱk+({|γk-1|>0})≠∅,thenIγ∉B(Hp,ϕS,Hq,ϕp/qS).
Assume that ℱk+({|γk-1|>0})≠∅. We fix B∈ℱk∖ℱk-1 and δ>0 such that P({|γk-1|≥δ}∩B)>0. Let B1={|γk-1|≥δ} and B1′={|γk-1|≥δ}∩B. Note that B1∈ℱk-1, B1′∈ℱk+({|γk-1|>0}), and B1′⊂B1.
To prove (122), we define two decreasing sequences of measurable sets {Bn}n=1∞ and {Bn′}n=1∞ that satisfy
(123)Bn∈ℱk-1,P(Bn)=P(B1)2n-1,Bn′∈ℱk+({|γk-1|>0}),Bn′⊂Bn
for every n≥1, inductively as follows.
Suppose that we can chooseBn-1 and Bn-1′ that satisfy
(124)Bn-1∈ℱk-1,P(Bn-1)=P(B1)2n-2,Bn-1′∈ℱk+({|γk-1|>0}),Bn-1′⊂Bn-1.
By the assumption that ℱ0 is nonatomic, ℱk-1 is also nonatomic. Hence, there exists Bn∈ℱk-1 such that Bn⊂Bn-1, P(Bn)=P(Bn-1)/2 and P(Bn∩Bn-1′)>0. Let Bn′=Bn∩Bn-1′. Then, we have Bn and Bn′ with Bn⊂Bn-1 and Bn′⊂Bn-1′ that satisfy (123).
For the setBn′ defined above, let gn=χBn′-Ek-1[χBn′]. Since Bn is ℱk-1-measurable, we have
(125)gnχΩ∖Bn=(χBn′-Ek-1[χBn′])χΩ∖Bn=χBn′χΩ∖Bn-Ek-1[χBn′χΩ∖Bn]=0.
By (125) and the assumption that t1/pϕ(t) is almost increasing, we have, for any A∈∪n=0∞ℱn,
(126)1ϕ(P(A))(1P(A)∫A‍|gn|pdP)1/p=1ϕ(P(A))P(A)1/p(∫A∩Bn‍|gn|pdP)1/p≲1ϕ(P(A∩Bn))P(A∩Bn)1/p×(∫A∩Bn‍|gn|pdP)1/p=1ϕ(P(A∩Bn))×(1P(A∩Bn)∫A∩Bn‍|gn|pdP)1/p.
We now show that Iγ∉B(Hp,ϕS,Hq,ϕp/qS). Suppose that Iγ∈B(Hp,ϕS,Hq,ϕp/qS); that is, there exists a positive number C such that
(127)∥Iγf∥Hq,ϕp/qS≤C∥f∥Hp,ϕS
for all f∈Hp,ϕS.
SinceBn′∈ℱk∖ℱk-1, we have dkgn=gn≠0, and djgn=0 for j≠k.
Therefore, we have(128)S(gn)=gn,S(Iγgn)=γk-1gn.
Forgn, we take Dn∈∪n=0∞ℱn such that
(129)1ϕ(P(Dn))(1P(Dn)∫Dn‍|gn|pdP)1/p≥12∥gn∥Hp,ϕS.
By (126), we may assume that Dn⊂Bn. As a consequence, we have Dn⊂B1={|γk-1|≥δ}, and limn→∞P(Dn)=0 by (123). Then, using (127) with (128), we have
(130)1ϕ(P(Dn))p/q(1P(Dn)∫Dn‍|gn|qdP)1/q≤1δ1ϕ(P(Dn))p/q×(1P(Dn)∫Dn‍|γk-1gn|qdP)1/q≤1δ∥Iγgn∥Hq,ϕp/qS≤Cδ∥gn∥Hp,ϕS≤2Cδ1ϕ(P(Dn))×(1P(Dn)∫Dn‍|gn|pdP)1/p≤2Cδ1ϕ(P(Dn))×(1P(Dn)∫Dn‍|gn|qdP)1/q.
Therefore, we have
(131)ϕ(P(Dn))1-p/q≤2Cδ.
However, this contradicts p<q and limt→0ϕ(t)=∞. We have (122) and hence have (119).



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*Source: 690258-2013-10-20.xml*