Martingale Morrey-Campanato Spaces and Fractional Integrals

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.

1. Introduction

The purpose of this paper is to introduce Morrey-Campanato spaces of martingales. The Lebesgue space 𝐿^𝑝 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(⋃_𝑛ℱ_𝑛), where {ℱ_𝑛}_𝑛≥0 is a nondecreasing sequence of sub-𝜎-algebras of ℱ. Following Weisz [5], we call {ℱ_𝑛}_𝑛≥0 a stochastic basis. For the sake of simplicity, let ℱ₋₁=ℱ₀. We suppose that every 𝜎-algebra ℱ_𝑛 is generated by countable atoms, where 𝐵∈ℱ_𝑛 is called an atom (more precisely a (ℱ_𝑛,𝑃)-atom), if any 𝐴⊂𝐵 with 𝐴∈ℱ_𝑛 satisfies 𝑃(𝐴)=𝑃(𝐵) or 𝑃(𝐴)=0. Denote by 𝐴(ℱ_𝑛) the set of all atoms in ℱ_𝑛. The expectation operator and the conditional expectation operators relative to ℱ_𝑛 are denoted by 𝐸 and 𝐸_𝑛, respectively.

We define Morrey-Campanato spaces as the following: let 𝑝∈[1,∞) and 𝜆∈(−∞,∞). For 𝑓∈𝐿₁, let‖𝑓‖_{𝐿𝑝,𝜆}=sup_𝑛≥0sup_{ℱ𝐵∈𝐴𝑛}1𝑃(𝐵)^𝜆𝑃1(𝐵)_𝐵‖𝑓‖_ℒ𝑝,𝜆||𝑓||^𝑝𝑑𝑃^1/𝑝,=sup_𝑛≥0sup_{ℱ𝐵∈𝐴𝑛}1𝑃(𝐵)^𝜆𝑃1(𝐵)_𝐵||𝑓−𝐸_𝑛𝑓||^𝑝𝑑𝑃^1/𝑝,(1.1) and let𝐿_𝑝,𝜆=𝑓∈𝐿^0𝑝∶‖𝑓‖_{𝐿𝑝,𝜆}<∞,ℒ_𝑝,𝜆=𝑓∈𝐿^0𝑝∶‖𝑓‖_ℒ𝑝,𝜆<∞,(1.2) where 𝐿^0𝑝 is the set of all 𝑓∈𝐿_𝑝 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 ℱ_𝑛 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 𝐿_𝑝-𝐿_𝑞 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 𝐿_𝑝, BMO, and Lip_𝛼 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 𝐿_𝑝 (𝐻_𝑝), 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 martingale 𝑓=(𝑓_𝑛)_𝑛≥0 relative to {ℱ_𝑛}_𝑛≥0, denote its martingale difference by 𝑑_𝑛𝑓=𝑓_𝑛−𝑓_𝑛−1 (𝑛≥0, with convention 𝑑₀𝑓=0). For 𝛼>0, we define the fractional integral 𝐼_𝛼𝑓=((𝐼_𝛼𝑓)_𝑛)_𝑛≥0 of 𝑓 by𝐼_𝛼𝑓_𝑛=^𝑛_𝑘=0𝑏^𝛼𝑘−1𝑑_𝑘𝑓,(1.3) where 𝑏_𝑘 is an ℱ_𝑘-measurable function such that𝑏_𝑘(𝜔)=𝑃(𝐵)fora.e.𝜔∈𝐵with𝐵∈𝐴ℱ_𝑘.(1.4) Then ((𝐼_𝛼𝑓)_𝑛)_𝑛≥0 is a martingale for any martingale 𝑓=(𝑓_𝑛)_𝑛≥0, since each 𝑏_𝑘 is bounded, that is, 𝐼_𝛼 is a martingale transform introduced by Burkholder [22]. This definition of 𝐼_𝛼 is an extention of the one in [20, 21] which is for dyadic martingales. We can prove the boundedness of fractional integrals 𝐼_𝛼 as martingale transforms from 𝐿_𝑝 to 𝐿_𝑞, if 1<𝑝<𝑞<∞ and −1/𝑝+𝛼=−1/𝑞. That is, if a martingale 𝑓=(𝑓_𝑛)_𝑛≥0 is 𝐿_𝑝-bounded, then ((𝐼_𝛼𝑓)_𝑛)_𝑛≥0 is 𝐿_𝑞-bounded and the following inequality holds:sup_𝑛≥0‖‖𝐼_𝛼𝑓_𝑛‖‖_𝐿𝑞≤𝐶sup_𝑛≥0‖‖𝑓_𝑛‖‖_𝐿𝑝,(1.5) where 𝐶 is a positive constant independent of 𝑓. Further, we prove the boundedness of fractional integrals 𝐼_𝛼 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 ℱ_𝑛 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 Section 3. 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 use 𝐶 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 𝐶_𝑝, 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 {ℱ_𝑛}_𝑛≥0 a nondecreasing sequence of sub-𝜎-algebras of ℱ such that ℱ=𝜎(⋃_𝑛ℱ_𝑛). For the sake of simplicity, let ℱ₋₁=ℱ₀. As in Section 1, we always suppose that every 𝜎-algebra ℱ_𝑛 is generated by countable atoms, with denoting by 𝐴(ℱ_𝑛) the set of all atoms in ℱ_𝑛. We define the fractional integral 𝐼_𝛼 as a martingale transform by (1.3).

For a martingale 𝑓=(𝑓_𝑛)_𝑛≥0 relative to {ℱ_𝑛}_𝑛≥0, the maximal function 𝑓^∗ of 𝑓 is defined by𝑓^∗𝑛=sup_{0≤𝑚≤𝑛}||𝑓_𝑚||,𝑓^∗=sup_𝑛≥0||𝑓_𝑛||.(2.1)

It is known that if 𝑝∈(1,∞), then any 𝐿_𝑝-bounded martingale converges in 𝐿_𝑝. Moreover, if 𝑓∈𝐿_𝑝, 𝑝∈[1,∞), then (𝑓_𝑛)_𝑛≥0 with 𝑓_𝑛=𝐸_𝑛𝑓 is an 𝐿_𝑝-bounded martingale and converges to 𝑓 in 𝐿_𝑝 (see, e.g., [26]). For this reason a function 𝑓∈𝐿₁ and the corresponding martingale (𝑓_𝑛)_𝑛≥0 with 𝑓_𝑛=𝐸_𝑛𝑓 will be denoted by the same symbol 𝑓. Note also that ‖𝑓‖_𝐿𝑝=sup_𝑛≥0‖𝐸_𝑛𝑓‖_𝐿𝑝. In this case𝑓^∗𝑛=sup_{0≤𝑚≤𝑛}||𝐸_𝑚𝑓||,𝑓^∗=sup_𝑛≥0||𝐸_𝑛𝑓||for𝑓∈𝐿₁.(2.2)

Let ℳ be the set of all martingales such that 𝑓₀=0. For 𝑝∈[1,∞], let 𝐿^0𝑝 be the set of all 𝑓∈𝐿_𝑝 such that 𝐸₀𝑓=0. For any 𝑓∈𝐿^0𝑝, let 𝑓_𝑛=𝐸_𝑛𝑓. Then (𝑓_𝑛)_𝑛≥0 is an 𝐿_𝑝-bounded martingale in ℳ. For this reason we can regard 𝐿^0𝑝 as a subset of ℳ.

In Section 1 we have introduced Morrey spaces 𝐿_𝑝,𝜆 and Campanato spaces ℒ_𝑝,𝜆 as the following.

Definition 2.1. Let 𝑝∈[1,∞) and 𝜆∈(−∞,∞). For 𝑓∈𝐿₁, let ‖𝑓‖_{𝐿𝑝,𝜆}=sup_𝑛≥0sup_{𝐵∈𝐴(ℱ𝑛)}1𝑃(𝐵)^𝜆𝑃1(𝐵)_𝐵‖𝑓‖_ℒ𝑝,𝜆||𝑓||^𝑝𝑑𝑃^1/𝑝,=sup_𝑛≥0sup_{ℱ𝐵∈𝐴𝑛}1𝑃(𝐵)^𝜆𝑃1(𝐵)_𝐵||𝑓−𝐸_𝑛𝑓||^𝑝𝑑𝑃^1/𝑝,(2.3) and define 𝐿_𝑝,𝜆=𝑓∈𝐿^0𝑝∶‖𝑓‖_{𝐿𝑝,𝜆}<∞,ℒ_𝑝,𝜆=𝑓∈𝐿^0𝑝∶‖𝑓‖_ℒ𝑝,𝜆<∞.(2.4)
Then functionals ‖𝑓‖_{𝐿𝑝,𝜆} and ‖𝑓‖_ℒ𝑝,𝜆 are norms on 𝐿_𝑝,𝜆 and ℒ_𝑝,𝜆, respectively. Note that 𝐿_𝑝,𝜆 and ℒ_𝑝,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 ℝ^𝑛.
The martingale 𝑓=(𝑓_𝑛)_𝑛≥0 is said to be 𝐿_𝑝,𝜆-bounded if 𝑓_𝑛∈𝐿_𝑝,𝜆 (𝑛≥0) and sup_𝑛≥0‖𝑓_𝑛‖_{𝐿𝑝,𝜆}<∞. Similarly, the martingale 𝑓=(𝑓_𝑛)_𝑛≥0 is said to be ℒ_𝑝,𝜆-bounded if 𝑓_𝑛∈ℒ_𝑝,𝜆 (𝑛≥0) and sup_𝑛≥0‖𝑓_𝑛‖_ℒ𝑝,𝜆<∞.

Proposition 2.2. Let 1≤𝑝<∞. Let 𝑓∈𝐿₁ and (𝑓_𝑛)_𝑛≥0 be its corresponding martingale with 𝑓_𝑛=𝐸_𝑛𝑓 (𝑛≥0).
(i)Assume that 𝜆∈(−∞,0]. If 𝑓∈𝐿_𝑝,𝜆, then (𝑓_𝑛)_𝑛≥0 is 𝐿_𝑝,𝜆-bounded and ‖𝑓‖_{𝐿𝑝,𝜆}≥sup_𝑛≥0‖‖𝑓_𝑛‖‖_{𝐿𝑝,𝜆}.(2.5) Conversely, if (𝑓_𝑛)_𝑛≥0 is 𝐿_𝑝,𝜆-bounded, then 𝑓∈𝐿_𝑝,𝜆 and ‖𝑓‖_{𝐿𝑝,𝜆}≤sup_𝑛≥0‖‖𝑓_𝑛‖‖_{𝐿𝑝,𝜆}.(2.6)(ii)Assume that 𝜆∈(−∞,∞). If 𝑓∈ℒ_𝑝,𝜆, then (𝑓_𝑛)_𝑛≥0 is ℒ_𝑝,𝜆-bounded and ‖𝑓‖_ℒ𝑝,𝜆≥sup_𝑛≥0‖‖𝑓_𝑛‖‖_ℒ𝑝,𝜆.(2.7) Conversely, if (𝑓_𝑛)_𝑛≥0 is ℒ_𝑝,𝜆-bounded, then 𝑓∈ℒ_𝑝,𝜆 and ‖𝑓‖_ℒ𝑝,𝜆≤sup_𝑛≥0‖‖𝑓_𝑛‖‖_ℒ𝑝,𝜆.(2.8)

Proof. (i) Let 𝑓∈𝐿_𝑝,𝜆 and 𝑛≥0. Fix any 𝐵∈𝐴(ℱ_𝑘). If 𝑘≤𝑛, then _𝐵||𝐸_𝑛𝑓||^𝑝𝑑𝑃^1/𝑝≤_𝐵𝐸_𝑛||𝑓||^𝑝𝑑𝑃^1/𝑝=_𝐵||𝑓||^𝑝𝑑𝑃^1/𝑝≤𝑃(𝐵)^𝜆+1/𝑝‖𝑓‖_{𝐿𝑝,𝜆}.(2.9) If 𝑛<𝑘, then taking 𝐵_𝑛∈𝐴(ℱ_𝑛) such that 𝐵⊂𝐵_𝑛, we have 𝐸_𝑛𝑓=1𝑃𝐵_𝑛^𝐵_𝑛𝑓𝑑𝑃on𝐵_𝑛.(2.10) Hence 𝑃1(𝐵)_𝐵||𝐸_𝑛𝑓||^𝑝𝑑𝑃^1/𝑝=1𝑃𝐵_𝑛^𝐵_𝑛||𝐸_𝑛𝑓||^𝑝𝑑𝑃^1/𝑝≤1𝑃𝐵_𝑛^𝐵_𝑛𝐸_𝑛||𝑓||^𝑝𝑑𝑃^1/𝑝=1𝑃𝐵_𝑛^𝐵_𝑛||𝑓||^𝑝𝑑𝑃^1/𝑝≤𝑃𝐵_𝑛^𝜆‖𝑓‖_{𝐿𝑝,𝜆}≤𝑃(𝐵)^𝜆‖𝑓‖_{𝐿𝑝,𝜆}.(2.11) Therefore, 𝑓_𝑛=𝐸_𝑛𝑓∈𝐿_𝑝,𝜆 and ‖𝑓_𝑛‖_{𝐿𝑝,𝜆}≤‖𝑓‖_{𝐿𝑝,𝜆} for all 𝑛≥0. This shows that (𝑓_𝑛)_𝑛≥0 is a 𝐿_𝑝,𝜆-bounded martingale and sup_𝑛≥0‖‖𝑓_𝑛‖‖_{𝐿𝑝,𝜆}≤‖𝑓‖_{𝐿𝑝,𝜆}.(2.12) Conversely, from the inequality _𝐵||𝑓||^𝑝𝑑𝑃≤liminf_𝑛→∞_𝐵||𝐸_𝑛𝑓||^𝑝𝑑𝑃∀𝐵∈_𝑚𝐴ℱ_𝑚,(2.13) it follows that ‖𝑓‖_{𝐿𝑝,𝜆}≤sup_𝑛≥0‖‖𝐸_𝑛𝑓‖‖_{𝐿𝑝,𝜆}.(2.14)(ii) Let 𝑓∈ℒ_𝑝,𝜆 and 𝑛≥0. Fix any 𝐵∈𝐴(ℱ_𝑘). If 𝑘≤𝑛, then_𝐵||𝐸_𝑛𝑓−𝐸_𝑘𝐸_𝑛𝑓||^𝑝𝑑𝑃^1/𝑝=_𝐵||𝐸_𝑛𝑓−𝐸_𝑛𝐸_𝑘𝑓||^𝑝𝑑𝑃^1/𝑝≤_𝐵𝐸_𝑛||𝑓−𝐸_𝑘𝑓||^𝑝𝑑𝑃^1/𝑝=_𝐵||𝑓−𝐸_𝑘𝑓||^𝑝𝑑𝑃^1/𝑝≤𝑃(𝐵)^𝜆+1/𝑝‖𝑓‖_ℒ𝑝,𝜆.(2.15) If 𝑛<𝑘, then 𝐸_𝑘[𝐸_𝑛𝑓]=𝐸_𝑛𝑓. Hence 𝑃1(𝐵)_𝐵||𝐸_𝑛𝑓−𝐸_𝑘𝐸_𝑛𝑓||^𝑝𝑑𝑃^1/𝑝=0.(2.16) Therefore, 𝑓_𝑛=𝐸_𝑛𝑓∈ℒ_𝑝,𝜆 and ‖𝑓_𝑛‖_ℒ𝑝,𝜆≤‖𝑓‖_ℒ𝑝,𝜆 for all 𝑛≥0. This shows that (𝑓_𝑛)_𝑛≥0 is a ℒ_𝑝,𝜆-bounded martingale and sup_𝑛≥0‖‖𝑓_𝑛‖‖_ℒ𝑝,𝜆≤‖𝑓‖_ℒ𝑝,𝜆.(2.17) Conversely, from the inequality _𝐵||𝑓−𝑓_𝑘||^𝑝𝑑𝑃≤liminf_𝑛→∞_𝐵||𝐸_𝑛𝑓−𝑓_𝑘||^𝑝𝑑𝑃∀𝐵∈𝐴ℱ_𝑘,𝑘=1,2,…,(2.18) it follows that ‖𝑓‖_ℒ𝑝,𝜆≤sup_𝑛≥0‖‖𝐸_𝑛𝑓‖‖_ℒ𝑝,𝜆.(2.19)

Remark 2.3. In general, for 𝑓∈𝐿_𝑝,𝜆 (resp., ℒ_𝑝,𝜆), 𝐸_𝑛𝑓 does not always converge to 𝑓 in 𝐿_𝑝,𝜆 (resp., ℒ_𝑝,𝜆). See Remark 3.7.

Remark 2.4. If ℱ₀={∅,Ω}, then 𝐿_𝑝,𝜆⊂𝐿_𝑝 and ‖𝑓‖_𝐿𝑝≤‖𝑓‖_{𝐿𝑝,𝜆} for 𝑓∈𝐿_𝑝,𝜆. Therefore, if (𝑓_𝑛)_𝑛≥0 is an 𝐿_𝑝,𝜆-bounded martingale, then it is an 𝐿_𝑝-bounded martingale. If 1<𝑝<∞, from the known result it follows that there exists 𝑓∈𝐿^0𝑝 such that 𝐸_𝑛𝑓=𝑓_𝑛, 𝑛≥0, and (𝑓_𝑛)_𝑛≥0 converges to 𝑓 in 𝐿_𝑝 and a.e. Moreover, we can deduce that 𝑓∈𝐿_𝑝,𝜆, since _𝐵||𝑓_𝑛(𝜔)||^𝑝𝑑𝑃^1/𝑝≤𝑃(𝐵)^𝜆+1/𝑝‖‖𝑓_𝑛‖‖_{𝐿𝑝,𝜆}≤𝑃(𝐵)^𝜆+1/𝑝sup_𝑛≥0‖‖𝑓_𝑛‖‖_{𝐿𝑝,𝜆},𝐵∈_𝑚𝐴ℱ_𝑚.(2.20) The stochastic basis {ℱ_𝑛}_𝑛≥0 is said to be regular, if there exists a constant 𝑅≥2 such that 𝑓_𝑛≤𝑅𝑓_𝑛−1(2.21) holds for all nonnegative martingales (𝑓_𝑛)_𝑛≥0.

Remark 2.5. In general, 𝐿_𝑝,𝜆⊂ℒ_𝑝,𝜆 with ‖𝑓‖_ℒ𝑝,𝜆≤2‖𝑓‖_{𝐿𝑝,𝜆}. Actually, for any 𝐵∈𝐴(ℱ_𝑛), 𝑃1(𝐵)_𝐵||𝑓−𝐸_𝑛𝑓||^𝑝𝑑𝑃^1/𝑝≤1𝑃(𝐵)_𝐵||𝑓||^𝑝𝑑𝑃^1/𝑝+1𝑃(𝐵)_𝐵||𝐸_𝑛𝑓||^𝑝𝑑𝑃^1/𝑝≤21𝑃(𝐵)_𝐵||𝑓||^𝑝𝑑𝑃^1/𝑝.(2.22) Moreover, if {ℱ_𝑛}_𝑛≥0 is regular and 𝜆<0, then we can prove that 𝐿_𝑝,𝜆=ℒ_𝑝,𝜆 with equivalent norms (Theorem 3.1 and Remark 3.2).

Remark 2.6. By definition, if 0≥𝜆≥𝜆^, then we have that 𝐿^0∞⊂𝐿_𝑝,𝜆⊂𝐿_𝑝,𝜆^′ with ‖𝑓‖_{𝐿′𝑝,𝜆}≤‖𝑓‖_{𝐿𝑝,𝜆}≤‖𝑓‖_𝐿∞ and 𝐿^0∞⊂ℒ_𝑝,𝜆⊂ℒ_𝑝,𝜆^′ with ‖𝑓‖_{ℒ′𝑝,𝜆}≤‖𝑓‖_ℒ𝑝,𝜆≤2‖𝑓‖_𝐿∞. If 𝜆≤−1/𝑝, then 𝐿^0𝑝⊂𝐿_𝑝,𝜆⊂ℒ_𝑝,𝜆 with ‖𝑓‖_ℒ𝑝,𝜆/2≤‖𝑓‖_{𝐿𝑝,𝜆}≤‖𝑓‖_𝐿𝑝.

Remark 2.7. By definition and Remark 2.5, if ℱ₀={∅,Ω}, then 𝐿_𝑝,𝜆⊂ℒ_𝑝,𝜆⊂𝐿^0𝑝 with ‖𝑓‖_𝐿𝑝≤‖𝑓‖_ℒ𝑝,𝜆≤2‖𝑓‖_{𝐿𝑝,𝜆}.

Definition 2.8. Let BMO=ℒ_1,0 and Lip_𝛼=ℒ_1,𝛼 if 𝛼>0.
Our definitions of BMO and Lip_𝛼 are different from the ones by Weisz [7]. To compare both we give another definition of martingale Morrey and Campanato spaces.

Definition 2.9. Let 𝑝∈[1,∞) and 𝜆∈(−∞,∞). For 𝑓∈𝐿₁, let ‖𝑓‖_{𝐿𝑝,𝜆,ℱ}=sup_𝑛≥0sup_{𝐵∈ℱ𝑛}1𝑃(𝐵)^𝜆𝑃1(𝐵)_𝐵‖𝑓‖_{ℒ𝑝,𝜆,ℱ}||𝑓||^𝑝𝑑𝑃^1/𝑝,=sup_𝑛≥0sup_{𝐵∈ℱ𝑛}1𝑃(𝐵)^𝜆𝑃1(𝐵)_𝐵||𝑓−𝐸_𝑛𝑓||^𝑝𝑑𝑃^1/𝑝,(2.23) and define𝐿_{𝑝,𝜆,ℱ}=𝑓∈𝐿^0𝑝∶‖𝑓‖_{𝐿𝑝,𝜆,ℱ}<∞,ℒ_{𝑝,𝜆,ℱ}=𝑓∈𝐿^0𝑝∶‖𝑓‖_{ℒ𝑝,𝜆,ℱ}<∞.(2.24)
Note that the spaces 𝐿_{𝑝,𝜆,ℱ} and ℒ_{𝑝,𝜆,ℱ} can be defined without the assumption that every 𝜎-algebra ℱ_𝑛 is generated by countable atoms.

Remark 2.10. By the definitions we have the relations 𝐿_{𝑝,𝜆,ℱ}⊂𝐿_𝑝,𝜆 with ‖𝑓‖_{𝐿𝑝,𝜆}≤‖𝑓‖_{𝐿𝑝,𝜆,ℱ} and ℒ_{𝑝,𝜆,ℱ}⊂ℒ_𝑝,𝜆 with ‖𝑓‖_ℒ𝑝,𝜆≤‖𝑓‖_{ℒ𝑝,𝜆,ℱ}. If 𝜆≥0, then we can prove that 𝐿_{𝑝,𝜆,ℱ}=𝐿_𝑝,𝜆 and ℒ_{𝑝,𝜆,ℱ}=ℒ_𝑝,𝜆 with the same norms, respectively (see Proposition 3.8). If −1/𝑝<𝜆<0, then 𝐿_{𝑝,𝜆,ℱ}⫋𝐿_𝑝,𝜆 and ℒ_{𝑝,𝜆,ℱ}⫋ℒ_𝑝,𝜆 in general (see Proposition 3.9).

Remark 2.11. It is known that, if {ℱ_𝑛}_𝑛≥0 is regular and 𝜆≥0, then ℒ_1,𝜆,ℱ=ℒ_{𝑝,𝜆,ℱ} with ‖𝑓‖_{ℒ1,𝜆,ℱ}≤‖𝑓‖_{ℒ𝑝,𝜆,ℱ}≤𝐶_𝑝‖𝑓‖_{ℒ1,𝜆,ℱ} for each 𝑝∈[1,∞) (see, e.g., [8]).
We also define weak Morrey spaces.

Definition 2.12. For 𝑝∈[1,∞) and 𝜆∈(−∞,∞), let ‖𝑓‖_{𝑊𝐿𝑝,𝜆}=sup_𝑛≥0sup_{𝐵∈𝐴(ℱ𝑛)}1𝑃(𝐵)^𝜆sup_𝑡>0𝑡𝑃𝐵∩||𝑓||>𝑡𝑃(𝐵)^1/𝑝,(2.25) for measurable functions 𝑓, and define 𝑊𝐿_𝑝,𝜆=𝑓∈𝐿⁰¹∶‖𝑓‖_{𝑊𝐿𝑝,𝜆}<∞.(2.26)

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 {ℱ_𝑛}_𝑛≥0 be regular, ℱ₀={∅,Ω} and (Ω,ℱ,𝑃) be nonatomic. Let 𝑝∈[1,∞). (i)If 𝜆≤−1/𝑝, then 𝐿_𝑝,𝜆=ℒ_𝑝,𝜆=𝐿^0𝑝 and 12‖𝑓‖^ℒ𝑝,𝜆≤‖𝑓‖_{𝐿𝑝,𝜆}=‖𝑓‖_𝐿𝑝≤‖𝑓‖_ℒ𝑝,𝜆.(3.1)(ii)If −1/𝑝<𝜆<0, then 𝐿^0∞⫋𝐿_𝑝,𝜆=ℒ_𝑝,𝜆⫋𝐿^0𝑝 and ‖𝑓‖_𝐿𝑝≤‖𝑓‖_{𝐿𝑝,𝜆}≤‖𝑓‖_𝐿∞,12‖𝑓‖^ℒ𝑝,𝜆≤‖𝑓‖_{𝐿𝑝,𝜆}≤𝐶‖𝑓‖_ℒ𝑝,𝜆.(3.2)(iii)If 𝜆=0, then 𝐿^0∞=𝐿_𝑝,0⫋ℒ_𝑝,0=BMO and ‖𝑓‖_𝐿𝑝,0=‖𝑓‖_𝐿∞,‖𝑓‖_BMO≤‖𝑓‖_ℒ𝑝,0≤𝐶_𝑝‖𝑓‖_𝐵𝑀𝑂.(3.3)(iv)If 𝜆>0, then {0}=𝐿_𝑝,𝜆⫋ℒ_𝑝,𝜆=Lip_𝜆 and ‖𝑓‖_Lip𝜆≤‖𝑓‖_ℒ𝑝,𝜆≤𝐶_𝑝‖𝑓‖_Lip𝜆.(3.4)

Remark 3.2. We can prove (i) without the assumption that {ℱ_𝑛}_𝑛≥0 is regular or that (Ω,ℱ,𝑃) is nonatomic. In (ii), we can prove that 𝐿_𝑝,𝜆=ℒ_𝑝,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∞⫋ℒ_𝑝,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 𝐿_𝑝,𝜆={0}.

To prove the theorem we first prove a lemma and two propositions.

Lemma 3.3. Let {ℱ_𝑛}_𝑛≥0 be regular. Then every sequence 𝐵₀⊃𝐵₁⊃⋯⊃𝐵_𝑛⊃⋯,𝐵_𝑛∈𝐴ℱ_𝑛,(3.5) has the following property: for each 𝑛≥1, 𝐵_𝑛=𝐵_𝑛−1or1+1𝑅𝑃𝐵^𝑛≤𝑃𝐵_𝑛−1≤𝑅𝑃𝐵_𝑛,(3.6) where 𝑅 is the constant in (2.21).

Remark 3.4. Since 𝐵_𝑛∈𝐴(ℱ_𝑛) is an (ℱ_𝑛,𝑃)-atom, we always interpret 𝐵_𝑛−1⊃𝐵_𝑛 as the inclusion modulo null sets, that is, 𝐵_𝑛−1⊃𝐵_𝑛 means 𝑃(𝐵_𝑛⧵𝐵_𝑛−1)=0. Therefore, 𝐵_𝑛=𝐵_𝑛−1 means 𝑃(𝐵_𝑛⧵𝐵_𝑛−1)=𝑃(𝐵_𝑛−1⧵𝐵_𝑛)=0.

Remark 3.5. By the lemma we see that there exists 𝑚 such that 𝐵_𝑚=𝐵_𝑛 for all 𝑛≥𝑚, if and only if lim_𝑛→∞𝑃(𝐵_𝑛)>0.

Proof of Lemma 3.3. Let 𝐵_𝑛−1={𝐸_𝑛−1[𝜒_𝐵𝑛]≥1/𝑅}. Then 𝐵_𝑛−1∈ℱ_𝑛−1. By the regularity we have 𝜒_𝐵𝑛≤𝑅𝐸_𝑛−1[𝜒_𝐵𝑛]. This shows 𝐵_𝑛−1⊃𝐵_𝑛. In this case, 𝐵_𝑛−1⊃𝐵_𝑛−1⊃𝐵_𝑛, since 𝐵_𝑛−1∈𝐴(ℱ_𝑛−1). From the definition of 𝐵_𝑛−1 it follows that 𝜒𝐵_𝑛−1≤𝑅𝐸_𝑛−1[𝜒_𝐵𝑛]. Then we have 𝑃𝐵_𝑛−1≤𝑃𝐵_𝑛−1=𝐸𝜒𝐵_𝑛−1≤𝐸𝑅𝐸^𝑛−1𝜒_𝐵𝑛=𝑅𝐸𝜒_𝐵𝑛=𝑅𝑃𝐵_𝑛.(3.7) Next we show 𝐵_𝑛−1=𝐵_𝑛 or (1+1/𝑅)𝑃(𝐵_𝑛)≤𝑃(𝐵_𝑛−1). Suppose that 𝑃𝐵_𝑛−1<1+1𝑅𝑃𝐵^𝑛.(3.8) Then 𝑃𝐵_𝑛−1⧵𝐵_𝑛=𝑃𝐵_𝑛−1−𝑃𝐵_𝑛<𝑃𝐵^𝑛𝑅≤𝑃𝐵^𝑛−1𝑅.(3.9) Therefore, 𝐸_𝑛−1𝜒_{𝐵𝑛−1⧵𝐵𝑛}=𝑃𝐵^𝑛−1⧵𝐵_𝑛𝑃𝐵_𝑛−1𝜒^𝐵𝑛−1<1𝑅𝜒^𝐵𝑛−1.(3.10) From the regularity and the inequality above it follows that 𝜒_{𝐵𝑛−1⧵𝐵𝑛}≤𝑅𝐸_𝑛−1𝜒_{𝐵𝑛−1⧵𝐵𝑛}<𝜒_𝐵𝑛−1.(3.11) This means that 𝐵_𝑛−1=𝐵_𝑛.

Proposition 3.6. Let {ℱ_𝑛}_𝑛≥0 be regular, 1≤𝑝<∞ and 𝜆>−1/𝑝.
(i)For a sequence 𝐵₀⊃𝐵₁⊃⋯⊃𝐵_𝑘⊃⋯, 𝐵_𝑘∈𝐴(ℱ_𝑘), let 𝑓₀=0 and 𝑓_𝑛=^𝑛_𝑘=1𝑃𝐵_𝑘^𝜆𝑃𝐵^𝑘−1𝑃𝐵_𝑘𝜒^𝐵𝑘−𝜒_𝐵𝑘−1,𝑛≥1.(3.12)Then 𝑓=(𝑓_𝑛)_𝑛≥0 is a martingale in ℳ and in ℒ_𝑝,𝜆.(ii)Let 0≥𝜆>𝜆′>−1/𝑝. If there exists a sequence 𝐵₀⊃𝐵₁⊃⋯⊃𝐵_𝑘⊃⋯, 𝐵_𝑘∈𝐴(ℱ_𝑘) and lim_𝑘→∞𝑃(𝐵_𝑘)=0, then 𝐿^0∞⫋ℒ_𝑝,𝜆⫋ℒ_𝑝,𝜆. If ℱ₀={∅,Ω} also, then 𝐿^0∞⫋ℒ_𝑝,𝜆⫋ℒ_𝑝,𝜆⫋𝐿^0𝑝.

Proof. (i) By the definition of the sequence (𝑓_𝑛)_𝑛≥0, we have 𝐸_𝑛−1𝑑_𝑛𝑓=𝑃𝐵_𝑛^𝜆𝑃𝐵^𝑛−1𝑃𝐵_𝑛𝐸^𝑛−1𝜒_𝐵𝑛−𝜒_𝐵𝑛−1=0,(3.13) for every 𝑛≥1. Hence, we obtain that (𝑓_𝑛)_𝑛≥0 is a martingale.
We next show that the sequence (𝑓_𝑛)_𝑛≥0 converges in 𝐿_𝑝. If lim_𝑘→∞𝑃(𝐵_𝑘)>0 then the convergence is clear by Remark 3.5. We assume that lim_𝑘→∞𝑃(𝐵_𝑘)=0. Then, by Lemma 3.3, we can take a sequence of integers 0=𝑘₀<𝑘₁<⋯<𝑘_𝑗<⋯ that satisfies 1+𝑅1𝑃𝐵^𝑘𝑗≤𝑃𝐵^𝑘_𝑗−1≤𝑅𝑃𝐵^𝑘_𝑗,(3.14) and 𝐵_𝑘𝑗−1=𝐵_𝑘 if 𝑘_𝑗−1≤𝑘<𝑘_𝑗. In this case we can write 𝑓_𝑛=_{𝑘𝑗≤𝑛}𝑃𝐵_𝑘𝑗^𝜆⎛⎜⎜⎝𝑃𝐵_𝑘𝑗−1𝑃𝐵_𝑘𝑗𝜒^𝐵𝑘𝑗−𝜒_{𝐵𝑘𝑗−1}⎞⎟⎟⎠.(3.15) Using (3.14) and the assumption 𝜆>−1/𝑝, we have _𝑘_𝑗>𝑛𝑃𝐵_𝑘𝑗^𝜆‖‖‖‖‖𝑃𝐵_𝑘𝑗−1𝑃𝐵_𝑘𝑗𝜒^𝐵𝑘𝑗−𝜒_{𝐵𝑘𝑗−1}‖‖‖‖‖_𝐿𝑝≤_{𝑘𝑗>𝑛}𝑃𝐵_𝑘𝑗^𝜆𝑅‖‖𝜒^𝐵𝑘𝑗‖‖_𝐿𝑝+‖‖𝜒_{𝐵𝑘𝑗−1}‖‖_𝐿𝑝≤2𝑅_{𝑘𝑗>𝑛}𝑃𝐵_𝑘𝑗^𝜆+1/𝑝≤2𝑅^∞_𝑗=01+𝑅1^{−(𝜆+1/𝑝)𝑗}𝑃𝐵_𝑛^𝜆+1/𝑝∼𝑃𝐵_𝑛^𝜆+1/𝑝.(3.16) Therefore, (𝑓_𝑛)_𝑛≥0 converges in 𝐿_𝑝. We denote by 𝑓 the limit of (𝑓_𝑛)_𝑛≥0.
We can also deduce from (3.16) that 𝑃1𝐵_𝑛^𝐵_𝑛||𝑓−𝐸_𝑛𝑓||^𝑝𝑑𝑃^1/𝑝≲𝑃𝐵_𝑛^𝜆.(3.17) On the other hand, for 𝐵∈𝐴(ℱ_𝑛), we have 𝑓−𝐸_𝑛𝑓𝜒_𝐵=𝑓−𝐸^𝑛0,𝑓,𝐵𝐵≠=𝐵𝐵_𝑛𝑛,.(3.18)
Combining (3.17) and (3.18), we have ‖𝑓‖_ℒ𝜆,𝑝≲1, that is, we get 𝑓∈ℒ_𝑝,𝜆.
(ii) First we show 𝐿^0∞⫋ℒ_𝑝,𝜆⫋ℒ_𝑝,𝜆^′. By Remark 2.6 we have 𝐿^0∞⊂ℒ_𝑝,𝜆⊂ℒ_𝑝,𝜆^′. Then we need to show ℒ_𝑝,0⧵𝐿^0∞≠∅ and ℒ_𝑝,𝜆^′⧵ℒ_𝑝,𝜆≠∅.
We consider 𝑓_𝑛 in (3.12) for the sequence 𝐵_𝑘, 𝑘=0,1,…. Then we can write 𝑓_𝑛=_{𝑘𝑗≤𝑛}⎛⎜⎜⎝𝑃𝐵_𝑘𝑗−1𝑃𝐵_𝑘𝑗𝜒^𝐵𝑘𝑗−𝜒_{𝐵𝑘𝑗−1}⎞⎟⎟⎠(3.19) for 𝜆=0 and we have 𝑓=(𝑓_𝑛)_𝑛≥0∈ℒ_𝑝,0. On the other hand, for a.e. 𝜔∈𝐵_𝑘𝑗⧵𝐵_𝑘𝑗+1, 𝑓=𝑓_𝑘𝑗+1=𝑓_𝑘𝑗−𝜒_𝐵𝑘𝑗=^𝑗_𝑖=0𝑃𝐵^𝑘𝑖−1𝑃𝐵_𝑘𝑖−1−1≥𝑅1(𝑗+1)−1.(3.20)
Then 𝑓∉𝐿^0∞ and ℒ_𝑝,0⧵𝐿^0∞≠∅.
Next, let 𝑔_𝑛=_{𝑘𝑗≤𝑛}𝑃𝐵_𝑘𝑗^𝜆⎛⎜⎜⎝𝑃𝐵_𝑘𝑗−1𝑃𝐵_𝑘𝑗𝜒^𝐵𝑘𝑗−𝜒_{𝐵𝑘𝑗−1}⎞⎟⎟⎠.(3.21) Then 𝑔=(𝑔_𝑛)_𝑛≥0∈ℒ_𝑝,𝜆^′. On the other hand, since |||𝑔−𝐸_𝑘𝑗−1𝑔|||=𝑃𝐵_𝑘𝑗^𝜆′on𝐵_𝑘𝑗−1⧵𝐵_𝑘𝑗,(3.22) we have 1𝑃𝐵_𝑘𝑗−1^𝜆⎛⎜⎜⎝1𝑃𝐵_𝑘𝑗−1^{𝐵𝑘𝑗−1}|||𝑔−𝐸_𝑘𝑗−1𝑔|||^𝑝𝑑𝑃⎞⎟⎟⎠^1/𝑝≥1𝑃𝐵_𝑘𝑗−1^𝜆⎛⎜⎜⎝1𝑃𝐵_𝑘𝑗−1^{𝐵𝑘𝑗−1⧵𝐵𝑘𝑗}|||𝑔−𝐸_𝑘𝑗−1𝑔|||^𝑝𝑑𝑃⎞⎟⎟⎠^1/𝑝≥𝑃𝐵^𝑘𝑗^𝜆′𝑃𝐵_𝑘𝑗−1^𝜆⎛⎜⎜⎝𝑃𝐵_𝑘𝑗−1⧵𝐵_𝑘𝑗𝑃𝐵_𝑘𝑗−1⎞⎟⎟⎠^1/𝑝≥𝑅^𝜆′(1+𝑅)^{−𝜆′−1/𝑝}𝑃𝐵_𝑘𝑗−1^{𝜆′−𝜆}⟶∞as𝑗⟶∞,(3.23) since 𝜆^<𝜆. This shows ℒ_𝑝,𝜆^′⧵ℒ_𝑝,𝜆≠∅.
Finally, if ℱ₀={∅,Ω} also, then by Remark 2.7 and ℒ_𝑝,𝜆^′⧵ℒ_𝑝,𝜆≠∅ we have ℒ_𝑝,𝜆⫋𝐿^0𝑝. This shows the conclusion.

Remark 3.7. In Proposition 3.6, 𝑓=(𝑓_𝑛)_𝑛≥0 in (3.12) converges in 𝐿_𝑝 as in the above proof. Moreover, the limit belongs to both ℒ_𝑝,𝜆 and 𝐿_𝑝,𝜆 when −1/𝑝<𝜆<0, since we will show that 𝐿_𝑝,𝜆=ℒ_𝑝,𝜆 in the proof of Theorem 3.1. However, it converges in neither ℒ_𝑝,𝜆 nor 𝐿_𝑝,𝜆. Actually, by a similar calculation to 𝑔=(𝑔_𝑛)_𝑛≥0 in the above proof, we have |||𝑓−𝑓_𝑛−𝐸_𝑘𝑗−1𝑓−𝑓_𝑛|||=|||𝑓−𝐸_𝑘𝑗−1𝑓|||=𝑃𝐵_𝑘𝑗^𝜆on𝐵_𝑘𝑗−1⧵𝐵_𝑘𝑗,(3.24) for 𝑛≤𝑘_𝑗−1, and then 1𝑃𝐵_𝑘𝑗−1^𝜆⎛⎜⎜⎝1𝑃𝐵_𝑘𝑗−1^{𝐵𝑘𝑗−1}|||𝑓−𝐸_𝑘𝑗−1𝑓|||^𝑝𝑑𝑃⎞⎟⎟⎠^1/𝑝≥𝑅^𝜆(1+𝑅)^{−𝜆−1/𝑝}.(3.25) By Remark 2.5, we have 2‖‖𝑓−𝑓_𝑛‖‖_{𝐿𝑝,𝜆}≥‖‖𝑓−𝑓_𝑛‖‖_ℒ𝑝,𝜆≥𝑅^𝜆(1+𝑅)^{−𝜆−1/𝑝}.(3.26) This shows that (𝑓_𝑛)_𝑛≥0 converges in neither ℒ_𝑝,𝜆 nor 𝐿_𝑝,𝜆.

Proposition 3.8. Let 1≤𝑝<∞.(i)If ℱ₀={∅,Ω} and 𝜆≤−1/𝑝, then 𝐿^0𝑝=𝐿_𝑝,𝜆=𝐿_{𝑝,𝜆,ℱ}=ℒ_𝑝,𝜆=ℒ_{𝑝,𝜆,ℱ} with‖𝑓‖_{𝐿𝑝,𝜆,ℱ}=‖𝑓‖_{𝐿𝑝,𝜆}=‖𝑓‖_𝐿𝑝≤‖𝑓‖_ℒ𝑝,𝜆≤‖𝑓‖_{ℒ𝑝,𝜆,ℱ}≤2‖𝑓‖_{𝐿𝑝,𝜆,ℱ}.(3.27)(ii)If 𝜆≥0, then 𝐿_𝑝,𝜆=𝐿_{𝑝,𝜆,ℱ}⊂ℒ_𝑝,𝜆=ℒ_{𝑝,𝜆,ℱ}⊂𝐿^0𝑝 with‖𝑓‖_𝐿𝑝≤‖𝑓‖_{𝐿𝑝,𝜆,ℱ}=‖𝑓‖_ℒ𝑝,𝜆≤‖𝑓‖_{𝐿𝑝,𝜆,ℱ}=‖𝑓‖_{𝐿𝑝,𝜆}.(3.28)

Proof. (i) Let 𝜆≤−1/𝑝. For any 𝑓∈𝐿^0𝑝 and any 𝐵∈ℱ_𝑛, 1𝑃(𝐵)^𝜆𝑃1(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.(3.29) 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) Let 𝜆≥0. By Remark 2.10 we need to show only 𝐿_𝑝,𝜆⊂𝐿_{𝑝,𝜆,ℱ} with ‖𝑓‖_{𝐿𝑝,𝜆,ℱ}≤‖𝑓‖_{𝐿𝑝,𝜆} and ℒ_𝑝,𝜆⊂ℒ_{𝑝,𝜆,ℱ}⊂𝐿^0𝑝 with ‖𝑓‖_𝐿𝑝≤‖𝑓‖_{ℒ𝑝,𝜆,ℱ}≤‖𝑓‖_ℒ𝑝,𝜆.
Let 𝑓∈𝐿_𝑝,𝜆. For any 𝐵∈ℱ_𝑛, there exists a sequence of atoms 𝐵_ℓ∈𝐴(ℱ_𝑛), ℓ=1,2,…, such that 𝐵=∪_ℓ𝐵_ℓ and 𝑃(𝐵)=∑_ℓ𝑃(𝐵_ℓ). Then _𝐵||𝑓||^𝑝𝑑𝑃=_ℓ_𝐵ℓ||𝑓||^𝑝𝑑𝑃≤_ℓ𝑃𝐵_ℓ^𝜆𝑝+1‖𝑓‖^𝑝𝐿_𝑝,𝜆≤𝑃𝐵_ℓ^𝜆𝑝+1‖𝑓‖^𝑝𝐿_𝑝,𝜆,(3.30) since 𝜆𝑝+1≥1. Therefore 𝑓∈𝐿_{𝑝,𝜆,ℱ} and ‖𝑓‖_{𝐿𝑝,𝜆,ℱ}≤‖𝑓‖_{𝐿𝑝,𝜆}. Similarly, we have ‖𝑓‖_{ℒ𝑝,𝜆,ℱ}≤‖𝑓‖_ℒ𝑝,𝜆 for 𝑓∈ℒ_𝑝,𝜆. By the definition of ℒ_{𝑝,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∈ℒ_{𝑝,𝜆,ℱ}.

Proof of Theorem 3.1. (i) We have the conclusion by Proposition 3.8 without the assumption that {ℱ_𝑛}_𝑛≥0 is regular or that (Ω,ℱ,𝑃) is nonatomic.
(ii) By Proposition 3.6 we only need to prove 𝐿_𝑝,𝜆=ℒ_𝑝,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.
Let 𝑓∈ℒ_𝑝,𝜆. Then, for any 𝐵∈𝐴(ℱ_𝑛), 𝑃(1𝐵)_𝐵|𝑓|^𝑝𝑑𝑃^1/𝑝≤1𝑃(𝐵)_𝐵|𝑓−𝐸_𝑛𝑓|^𝑝𝑑𝑃^1/𝑝+||||1𝑃(𝐵)_𝐵≤𝑃(𝐵)_𝜆𝑓(𝜔)𝑑𝑃||||‖𝑓‖_ℒ𝑝,𝜆+||||1𝑃(𝐵)_𝐵𝑓(𝜔)𝑑𝑃||||,(3.31) since 𝐸_𝑛𝑓=1𝑃(𝐵)_𝐵𝑓(𝜔)𝑑𝑃on𝐵.(3.32) If 𝐵∈ℱ₀, then 𝐸_𝑛𝑓=0 on 𝐵. Assume that 𝐵∉ℱ₀. By Lemma 3.3 we can choose 𝐵_𝑘𝑗∈𝐴(ℱ_𝑘𝑗), 0=𝑘₀<𝑘₁<⋯<𝑘_𝑚≤𝑛, such that 𝐵_𝑘0⊃𝐵_𝑘1⊃𝐵_𝑘2⊃⋯⊃𝐵_𝑘𝑚=𝐵 and that (1+1/𝑅)𝑃(𝐵_𝑘𝑗)≤𝑃(𝐵_𝑘𝑗−1)≤𝑅𝑃(𝐵_𝑘𝑗). Then, since 1𝑃𝐵_𝑘0^𝐵_𝑘0𝑓(𝜔)𝑑𝑃=0,(3.33) we have 1𝑃(𝐵)_𝐵𝑓(𝜔)𝑑𝑃=1𝑃(𝐵)_𝐵𝑓(𝜔)𝑑𝑃−1𝑃𝐵_𝑘0^𝐵_𝑘0=𝑓(𝜔)𝑑𝑃_𝑚_𝑗=1⎛⎜⎜⎝1𝑃𝐵_𝑘𝑗^𝐵𝑘𝑗𝑓(𝜔)𝑑𝑃−1𝑃𝐵_𝑘𝑗−1^{𝐵𝑘𝑗−1}=𝑓(𝜔)𝑑𝑃_𝑚⎞⎟⎟⎠_𝑗=11𝑃𝐵_𝑘𝑗^𝐵𝑘𝑗𝑓−𝐸^𝑘_𝑗−1𝑓(𝜔)𝑑𝑃.(3.34) By Hölder’s inequality and the assumption 𝜆<0 we have ||||1𝑃(𝐵)_𝐵𝑓(𝜔)𝑑𝑃||||≤^𝑚_𝑗=1⎛⎜⎜⎝1𝑃𝐵_𝑘𝑗^𝐵𝑘𝑗|||𝑓−𝐸_𝑘𝑗−1𝑓|||^𝑝𝑑𝑃⎞⎟⎟⎠^1/𝑝≲^𝑚_𝑗=1⎛⎜⎜⎝1𝑃𝐵_𝑘𝑗−1^{𝐵𝑘𝑗−1}|||𝑓−𝐸_𝑘𝑗−1𝑓|||^𝑝𝑑𝑃⎞⎟⎟⎠^1/𝑝≤^𝑚_𝑗=1𝑃𝐵_𝑘𝑗−1^𝜆‖𝑓‖_ℒ𝑝,𝜆≤^𝑚_𝑗=11+R1^{𝑚−𝑗+1}𝑃𝐵_𝑘𝑚^𝜆‖𝑓‖_ℒ𝑝,𝜆∼𝑃(𝐵)^𝜆‖𝑓‖_ℒ𝑝,𝜆.(3.35) Therefore we have ‖𝑓‖_{𝐿𝑝,𝜆}≲‖𝑓‖_ℒ𝑝,𝜆.
(iii) Let 𝜆=0. By Remark 2.6 and Proposition 3.6 we have 𝐿^0∞⊂𝐿_𝑝,0 with ‖𝑓‖_𝐿𝑝,0≤‖𝑓‖_𝐿∞ and 𝐿^0∞⫋ℒ_𝑝,0.
Next we show 𝐿_𝑝,0⊂𝐿^0∞ and ‖𝑓‖_𝐿∞≤‖𝑓‖_𝐿𝑝,0.
Let 𝑓∈𝐿_𝑝,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∈ℱ_𝑛 such that 𝑃𝐵∩||𝑓||>𝑟>(1−𝜖)𝑃(𝐵),(3.36) because ℱ is generated by ∪_𝑛ℱ_𝑛. For the above 𝐵, we can take a sequence of atoms 𝐵_ℓ∈𝐴(ℱ_𝑛), ℓ=1,2,…, such that 𝐵=∪_ℓ𝐵_ℓ and 𝑃(𝐵)=∑_ℓ𝑃(𝐵_ℓ). Hence, by the pigeonhole principle, there exists 𝐵′∈𝐴(ℱ_𝑛) such that 𝑃𝐵^∩||𝑓||>𝑟>(1−𝜖)𝑃𝐵^.(3.37) Therefore, we have ‖𝑓‖^𝑝𝐿_𝑝,0≥1𝑃𝐵^^𝐵′≥1||𝑓||^𝑝𝑑𝑃𝑃𝐵^^{𝐵∩{|𝑓|>𝑟}}≥𝑃𝐵||𝑓||^𝑝𝑑𝑃_∩||𝑓||>𝑟𝑃𝐵^𝑟^𝑝≥(1−𝜖)𝑟^𝑝.(3.38) This shows that 𝑃(|𝑓|>𝑟)>0 implies ‖𝑓‖_𝐿𝑝,0≥𝑟. Then we have the conclusion.
By Proposition 3.8 and Remark 2.11 we have that BMO(=ℒ_1,𝜆)=ℒ_𝑝,𝜆 with equivalent norms.
(iv) Let 𝜆>0. For 𝑓∈𝐿_𝑝,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(ℱ_𝑛+1) and we have the same estimate as (3.37) for some atom in 𝐴(ℱ_𝑛+1). Therefore, we can take an atom 𝐵′ in 𝐴(ℱ_𝑛+𝑚) for large enough 𝑚 such that 𝐵′ satisfies (3.37) and 𝑃(𝐵′)<𝜖. Hence we have ‖𝑓‖^𝑝𝐿_𝑝,𝜆≥1𝑃𝐵^^1+𝜆𝑝_𝐵||𝑓||^𝑝𝑑𝑃≥(1−𝜖)𝑟^𝑝𝑃𝐵^^𝜆𝑝≥(1−𝜖)𝑟^𝑝𝜀^𝜆𝑝,(3.39) that is, 𝑟≤𝜖^𝜆/(1−𝜖)^1/𝑝‖𝑓‖_{𝐿𝑝,𝜆}. This contradicts that 𝑟>0. Then 𝑓=0 a.e. and 𝐿_𝑝,𝜆={0}. By Proposition 3.6 we have that {0}⫋ℒ_𝑝,𝜆.
Finally, by Proposition 3.8 and Remark 2.11 we have that ℒ_1,𝜆=ℒ_𝑝,𝜆 with equivalent norms.

Next we prove that 𝐿_{𝑝,𝜆,ℱ}⫋𝐿_𝑝,𝜆 and ℒ_{𝑝,𝜆,ℱ}⫋ℒ_𝑝,𝜆 in general by an example.

Proposition 3.9. Let (Ω,ℱ,𝑃) be as follows: ℱΩ=[0,1),𝐴ℱ_𝑛=𝐼_𝑛,𝑗=𝑗2^−𝑛,(𝑗+1)2^−𝑛)∶𝑗=0,1,…,2^𝑛−1,(3.40)_𝑛=𝜎𝐴ℱ_𝑛,ℱ=𝜎_𝑛ℱ_𝑛,𝑃=theLebesguemeasure.(3.41) If −1/𝑝<𝜆<0, then 𝐿_{𝑝,𝜆,ℱ}⫋𝐿_𝑝,𝜆 and ℒ_{𝑝,𝜆,ℱ}⫋ℒ_𝑝,𝜆.

Proof. We construct 𝑓 such that 𝑓∈𝐿_𝑝,𝜆⧵𝐿_{𝑝,𝜆,ℱ},𝑓∈ℒ_𝑝,𝜆⧵ℒ_{𝑝,𝜆,ℱ}.(3.42)
Step 1. Denote the characteristic function of 𝐼_𝑛,𝑗 by 𝜒_𝑛,𝑗 and let 𝑓_𝑛=^2𝑛−1_𝑗=0𝑓_𝑛+𝑚,2^𝑚_𝑗,𝑓_𝑛,𝑗=𝑃𝐼_𝑛,𝑗^𝜆𝜒_𝑛+1,2𝑗−𝜒_{𝑛+1,2𝑗+1},(3.43) where we choose 𝑚 such that 𝑃𝐼_𝑛+𝑚,0^𝑝𝜆+1≤𝑃𝐼_𝑛,0.(3.44)
Note that 𝐼_𝑛,𝑗=𝐼_𝑛+1,2𝑗∪𝐼_{𝑛+1,2𝑗+1} and |𝑓_𝑛,𝑗|=𝑃(𝐼_𝑛,0)^𝜆𝜒_𝑛,𝑗.
If 𝑘≤𝑛+𝑚, then 𝐸_𝑘𝑓_𝑛=0, and |𝑓_𝑛−𝐸_𝑘𝑓_𝑛|=|𝑓_𝑛|=𝑃(𝐼_𝑛+𝑚,0)^𝜆∑^{2𝑛−1𝑗=1}𝜒_𝑛+𝑚,2^𝑚_𝑗. Hence𝑃𝐼1_𝑘,ℓ^𝐼_𝑘,ℓ||𝑓_𝑛−𝐸_𝑘𝑓_𝑛||^𝑝𝑑𝑃^1/𝑝=1𝑃(𝐼_𝑘,ℓ)_𝐼𝑘,ℓ||𝑓_𝑛||^𝑝𝑑𝑃^1/𝑝=𝑃𝐼_𝑛+𝑚,0^𝜆∑^{𝑗∶𝐼𝑚𝑗𝑛+𝑚,2⊂𝐼𝑘,ℓ}𝑃𝐼_𝑛+𝑚,2^𝑚_𝑗𝑃𝐼_𝑘,ℓ^1/𝑝.(3.45)
If 𝑘≤𝑛, then the number of the elements of {𝑗∶𝐼_𝑛+𝑚,2^𝑚_𝑗⊂𝐼_𝑘,ℓ} is the same as of {𝑗∶𝐼_𝑛,𝑗⊂𝐼_𝑘,ℓ}. Hence 𝑃(𝐼1_𝑘,ℓ)_𝐼𝑘,ℓ||𝑓_𝑛−𝐸_𝑘𝑓_𝑛||^𝑝𝑑𝑃^1/𝑝=1𝑃𝐼_𝑘,ℓ^𝐼_𝑘,ℓ||𝑓_𝑛||^𝑝𝑑𝑃^1/𝑝≤𝑃𝐼_𝑛+𝑚,0^𝜆𝑃𝐼^𝑛+𝑚,0𝑃𝐼_𝑛,0^1/𝑝≤1≤𝑃𝐼_𝑘,ℓ^𝜆,(3.46) 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∶𝐼_𝑛+𝑚,2^𝑚_𝑗⊂𝐼_𝑘,ℓ} is one at most. Hence 𝑃𝐼1_𝑘,ℓ^𝐼_𝑘,ℓ||𝑓_𝑛−𝐸_𝑘𝑓_𝑛||^𝑝𝑑𝑃^1/𝑝=1𝑃𝐼_𝑘,ℓ^𝐼_𝑘,ℓ||𝑓_𝑛||^𝑝𝑑𝑃^1/𝑝≤𝑃𝐼_𝑛+𝑚,0^𝜆𝑃𝐼^𝑛+𝑚,0𝑃𝐼_𝑘,ℓ^1/𝑝≤𝑃𝐼_𝑘,ℓ^𝜆,(3.47) where we use −1/𝑞<𝜆<0 for the last inequality. In the above, if 𝐼_𝑘,ℓ=𝐼_𝑛+𝑚,0, then the equality holds.
If 𝑘>𝑛+𝑚, then 𝑓_𝑛−𝐸_𝑘𝑓_𝑛=0 and 𝑃𝐼1_𝑘,ℓ^𝐼_𝑘,ℓ||𝑓_𝑛||^𝑝𝑑𝑃^1/𝑝≤𝑃𝐼_𝑛+𝑚,0^𝜆≤𝑃𝐼_𝑘,0^𝜆.(3.48) Therefore, 𝑓_𝑛∈𝐿_𝑝,𝜆∩ℒ_𝑝,𝜆,‖‖𝑓_𝑛‖‖_{𝐿𝑝,𝜆}=‖‖𝑓_𝑛‖‖_ℒ𝑝,𝜆=1.(3.49) On the other hand, for the set 𝐵=⋃^{2𝑛−1𝑗=0}𝐼_𝑛+𝑚,2^𝑚_𝑗∈ℱ_𝑛+𝑚, 𝑃1(𝐵)_𝐵||𝑓_𝑛−𝐸_𝑛+𝑚𝑓_𝑛||^𝑝𝑑𝑃^1/𝑝=1𝑃(𝐵)_𝐵||𝑓_𝑛||^𝑝𝑑𝑃^1/𝑝=𝑃𝐼_𝑛+𝑚,0^𝜆=(2^−𝑛𝑃(𝐵))^𝜆.(3.50)
Therefore, ‖𝑓_𝑛‖_{𝐿𝑝,𝜆,ℱ},‖𝑓_𝑛‖_{ℒ𝑝,𝜆,ℱ}≥2^−𝑛𝜆→∞ as 𝑛→∞. Step 2. Let 𝑓_𝑛 be as in Step 1. If 𝑘<𝑛, then, by the same observation as Step 1 we also have that 𝑓_𝑛𝜒_𝑘,ℓ∈𝐿_𝑝,𝜆∩ℒ_𝑝,𝜆,‖‖𝑓_𝑛𝜒_𝑘,ℓ‖‖_{𝐿𝑝,𝜆}=‖‖𝑓_𝑛𝜒_𝑘,ℓ‖‖_ℒ𝑝,𝜆=1.(3.51) Moreover, if 𝑘<𝑛, then, for the set 𝐵=(⋃^{2𝑛−1𝑗=0}𝐼_𝑛+𝑚,2^𝑚_𝑗)∩𝐼_𝑘,ℓ∈ℱ_𝑛+𝑚, 𝑃1(𝐵)_𝐵||𝑓_𝑛𝜒_𝑘,ℓ−𝐸_𝑛+𝑚𝑓_𝑛𝜒_𝑘,ℓ||^𝑝𝑑𝑃^1/𝑝=1𝑃(𝐵)_𝐵||𝑓_𝑛𝜒_𝑘,ℓ||^𝑝𝑑𝑃^1/𝑝=𝑃𝐼_𝑛+𝑚,0^𝜆=2^−𝑛+𝑘𝑃(𝐵)^𝜆.(3.52) Therefore, ‖𝑓_𝑛𝜒_𝑘,ℓ‖_{𝐿𝑝,𝜆,ℱ},‖𝑓_𝑛𝜒_𝑘,ℓ‖_{ℒ𝑝,𝜆,ℱ}≥2^{(−𝑛+𝑘)𝜆}.Step 3. Let 𝑓_𝑛 be as in Step 1 and let 𝑓=^∞_𝑛=12^𝑛𝜆/2𝑓_2𝑛𝜒_𝑛,1.(3.53) Note that ∑^∞𝑛=1𝜒_𝑛,1≡1. Then ‖𝑓‖_{𝐿𝑝,𝜆},‖𝑓‖_ℒ𝑝,𝜆≤^∞_𝑛=12^𝑛𝜆/2=2^𝜆/21−2^𝜆/2.(3.54) On the other hand, by the same observation as Step 2, we have that ‖𝑓‖_{𝐿𝑝,𝜆,ℱ},‖𝑓‖_{ℒ𝑝,𝜆,ℱ}≥2^𝑛𝜆/2×2^{(−2𝑛+𝑛)𝜆}=2^{−𝑛𝜆/2},(3.55) 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‖𝑓‖_𝑊𝐿𝑝=sup_𝑡>0𝑡𝑃||𝑓||>𝑡^1/𝑝.(3.56)

The following is well known for classical Morrey spaces on ℝ^𝑛. We give a proof for convenience, though it is the same as the proof for classical Morrey spaces on ℝ^𝑛.

Proposition 3.10. If 1<𝑞<𝑝<∞ and −1/𝑝=𝜆, then ‖𝑓‖_𝐿1,𝜆≤‖𝑓‖_{𝐿𝑞,𝜆}≤𝐶‖𝑓‖_𝑊𝐿𝑝.(3.57)

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(ℱ_𝑛), let 𝜂=𝑃(𝐵)^𝜆=𝑃(𝐵)^−1/𝑝 and 𝑓=𝑓^𝜂+𝑓_𝜂,𝑓^𝜂(𝜔)=𝑓0,(𝜔),||||𝑓𝑓((𝜔𝜔))||||>≤𝜂𝜂,.(3.58) Then 1𝑃(𝐵)^𝑞𝜆+1_𝐵||𝑓^𝜂(𝜔)||^𝑞𝑑𝑃≤1𝑃(𝐵)^𝑞𝜆+1^∞0≤1𝑃||𝑓^𝜂||>𝑡𝑞𝑡^𝑞−1𝑑𝑡𝑃(𝐵)^𝑞𝜆+1^𝜂0≤1𝑃||𝑓^𝜂||>𝜂𝑞𝑡^𝑞−1𝑑𝑡+^∞𝜂𝑡^−𝑝𝑞𝑡^𝑞−1𝑑𝑡𝑃(𝐵)^𝑞𝜆+11+𝑝−𝑞𝑞𝜂^𝑞−𝑝=𝑝𝑝−𝑞.(3.59) Since |𝑓_𝜂(𝜔)|≤𝜂 we have 1𝑃(𝐵)^𝜆𝑃1(𝐵)_𝐵||𝑓_𝜂(𝜔)||^𝑞𝑑𝑃^1/𝑞≤𝜂𝑃(𝐵)^𝜆=1.(3.60) We get the conclusion.

4. Maximal Function

It is known as Doob’s inequality that (see for example [5, Pages 20-21])‖𝑓^∗‖_𝐿𝑝≤𝑝𝑝−1‖𝑓‖^𝐿𝑝‖𝑓_∗,𝑓∈𝐿_𝑝(𝑝>1),(4.1)‖_𝑊𝐿1≤‖𝑓‖_𝐿1,𝑓∈𝐿₁.(4.2)

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 {ℱ_𝑛}_𝑛≥0.

Theorem 4.1. Let 1≤𝑝<∞ and −1/𝑝≤𝜆<0. Then, for 𝑓∈𝐿₁, ‖‖𝑓𝑓^∗_∗‖_{𝐿𝑝,𝜆}≤𝐶_𝑝‖𝑓‖_{𝐿𝑝,𝜆},if𝑝>1,‖_{𝑊𝐿1,𝜆}≤𝐶₁‖𝑓‖_𝐿1,𝜆,if𝑝=1.(4.3)

Proof. Case 1 (𝑝>1). For any 𝐵∈𝐴(ℱ_𝑚) and 𝑚≥0, let 𝑓=𝑔+ℎ and 𝑔=𝑓𝜒_𝐵. Then, using (4.1), we have _𝐵𝑔^∗^𝑝𝑑𝑃≤_Ω𝑔^∗^𝑝𝑑𝑃≲_Ω||𝑔||^𝑝𝑑𝑃=_𝐵||𝑓||^𝑝𝑑𝑃.(4.4) Hence 1𝑃(𝐵)^𝜆𝑃1(𝐵)_𝐵𝑔^∗^𝑝𝑑𝑃^1/𝑝≲‖𝑓‖_{𝐿𝑝,𝜆}.(4.5)
Next, take 𝐵_𝑛∈𝐴(ℱ_𝑛), 𝑛=0,1,…,𝑚, such that 𝐵=𝐵_𝑚⊂𝐵_𝑚−1⊂…⊂𝐵₀. Then, for a.e. 𝜔∈𝐵, 𝐸_𝑛ℎ(𝜔)=⎧⎪⎨⎪⎩0,1(𝑛≥𝑚),𝑃𝐵_𝑛^𝐵_𝑛ℎ𝑑𝑃,(𝑛<𝑚).(4.6) If 𝑛<𝑚, then ||𝐸_𝑛ℎ(𝜔)||≤1𝑃𝐵_𝑛^𝐵_𝑛||ℎ||^𝑝𝑑𝑃^1/𝑝≤𝑃𝐵_𝑛^𝜆‖𝑓‖_{𝐿𝑝,𝜆}≤𝑃(𝐵)^𝜆‖𝑓‖_{𝐿𝑝,𝜆},(4.7) since 𝜆<0. Hence ℎ^∗≤𝑃(𝐵)^𝜆‖𝑓‖_{𝐿𝑝,𝜆}on𝐵.(4.8)
Then 1𝑃(𝐵)^𝜆𝑃1(𝐵)_𝐵ℎ^∗^𝑝𝑑𝑃^1/𝑝≤‖𝑓‖_{𝐿𝑝,𝜆}.(4.9)
By (4.5), (4.9), and the inequality 𝑓^∗≤𝑔^∗+ℎ^∗, we have 1𝑃(𝐵)^𝜆𝑃1(𝐵)_𝐵𝑓^∗^𝑝𝑑𝑃^1/𝑝≲‖𝑓‖_{𝐿𝑝,𝜆},(4.10) 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, 𝑡𝑃𝐵∩𝑔^∗>𝑡≤𝑡𝑃𝑔^∗>𝑡≤_Ω||𝑔||𝑑𝑃=_𝐵||𝑓||𝑑𝑃(4.11) and then 1𝑃(𝐵)^𝜆𝑡𝑃𝐵∩𝑔^∗>𝑡𝑃(𝐵)≤‖𝑓‖^𝐿1,𝜆.(4.12) We also have (4.8) for the case 𝑝=1. Then 1𝑃(𝐵)^𝜆𝑡𝑃𝐵∩ℎ^∗>𝑡𝑃(𝐵)≤‖𝑓‖^𝐿1,𝜆.(4.13) 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 of 𝐼_𝛼 as a martingale transform.

For normed spaces 𝑀₁ and 𝑀₂ of functions, we denote by 𝐵(𝑀₁,𝑀₂) the set of all bounded martingale transforms from 𝑀₁ to 𝑀₂, that is, 𝐼_𝛼∈𝐵(𝑀₁,𝑀₂) means thatsup_𝑛≥0‖‖𝐼_𝛼𝑓_𝑛‖‖_𝑀2≤𝐶sup_𝑛≥0‖‖𝑓_𝑛‖‖_𝑀1,(5.1) for all 𝑀₁-bounded martingales 𝑓=(𝑓_𝑛)_𝑛≥0.

We state our results in Section 5.1. To prove Theorems 5.1 and 5.5 we show the pointwise estimate for 𝐼_𝛼𝑓 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 for 𝐿_𝑝-𝐿_𝑞 boundedness.

Theorem 5.1. Assume that {ℱ_𝑛}_𝑛≥0 is regular. Let 1≤𝑝<𝑞<∞, −1/𝑝+𝛼=−1/𝑞. Then, for 𝑓∈𝐿₁, ‖‖‖‖𝐼𝐼_𝛼𝑓^∗‖‖_𝐿𝑞≤𝐶‖𝑓‖_𝐿𝑝,if𝑝>1,(5.2)_𝛼𝑓^∗‖‖_𝑊𝐿𝑞≤𝐶‖𝑓‖_𝐿1,if𝑝=1.(5.3)

Remark 5.2. Let 𝑓=(𝑓_𝑛)_𝑛≥0 be an 𝐿₁-bounded martingale. For each 𝐿₁-function 𝑓_𝑚, 𝑚≥0, consider the corresponding martingale 𝑓^𝑚=(𝐸_𝑛[𝑓_𝑚])_𝑛≥0=(𝑓_{min(𝑛,𝑚)})_𝑛≥0. Then (𝐼_𝛼𝑓^𝑚)_𝑛=(𝐼_𝛼𝑓)_𝑛 for 𝑚≥𝑛. Therefore, from (5.3) it follows that, for 𝑚≥𝑛, ‖‖𝐼_𝛼𝑓_𝑛‖‖_𝑊𝐿𝑞=‖‖𝐼_𝛼𝑓^𝑚_𝑛‖‖_𝑊𝐿𝑞≲‖‖𝑓_𝑚‖‖_𝐿1≤sup_𝑛≥0‖‖𝑓_𝑛‖‖_𝐿1.(5.4) This shows that sup_𝑛≥0‖‖𝐼_𝛼𝑓_𝑛‖‖_𝑊𝐿𝑞≤𝐶sup_𝑛≥0‖‖𝑓_𝑛‖‖_𝐿1.(5.5)

Remark 5.3. Let a martingale 𝑓=(𝑓_𝑛)_𝑛≥0 be 𝐿₁-bounded. Since 𝑊𝐿_𝑞⊂𝐿_𝑞1 and ‖(𝐼_𝛼𝑓)_𝑛‖_𝐿𝑞1≤‖(𝐼_𝛼𝑓)_𝑛‖_𝑊𝐿𝑞 for 1<𝑞₁<𝑞, from Remark 5.2 it follows that the martingale ((𝐼_𝛼𝑓)_𝑛)_𝑛≥0 is 𝐿_𝑞1-bounded and that it converges in 𝐿_𝑞1. Denote this limit by 𝐼_𝛼𝑓. Then, 𝐼_𝛼𝑓=∑^∞𝑘=0𝑏^𝛼𝑘−1𝑑_𝑘𝑓 and 𝐸_𝑛[𝐼_𝛼𝑓]=(𝐼_𝛼𝑓)_𝑛.
If 𝑝>1, then the same observation as in Remark 5.2 with (5.2) shows that sup_𝑛≥0‖‖𝐼_𝛼𝑓_𝑛‖‖_𝐿𝑞≤𝐶sup_𝑛≥0‖‖𝑓_𝑛‖‖_𝐿𝑝.(5.6) Hence we have the following corollary.

Corollary 5.4. Assume that {ℱ_𝑛}_𝑛≥0 is regular. Let 1≤𝑝<𝑞<∞, −1/𝑝+𝛼=−1/𝑞. Then 𝐼𝐼_𝛼∈𝐵𝐿_𝑝,𝐿_𝑞,if𝑝>1,_𝛼∈𝐵𝐿₁,𝑊𝐿_𝑞,if𝑝=1.(5.7)

For Morrey norms, one has the following.

Theorem 5.5. Assume that {ℱ_𝑛}_𝑛≥0 is regular. Let 1≤𝑝<∞, −1/𝑝≤𝜆<0, and 𝜇=𝜆+𝛼<0. Then, for 𝑓∈𝐿₁, ‖‖‖‖𝐼𝐼_𝛼𝑓^∗‖‖_{𝐿𝑞,𝜇}≤𝐶‖𝑓‖_{𝐿𝑝,𝜆},if𝑝>1,1≤𝑞≤(𝜆/𝜇)𝑝,(5.8)_𝛼‖‖𝐼𝑓^∗‖‖_{𝐿𝑞,𝜇}≤𝐶‖𝑓‖_𝐿1,𝜆,if𝑝=1,1≤𝑞<(𝜆/𝜇)𝑝,(5.9)_𝛼𝑓^∗‖‖_{𝑊𝐿𝑞,𝜇}≤𝐶‖𝑓‖_𝐿1,𝜆,if𝑝=1,𝑞=(𝜆/𝜇)𝑝.(5.10)

Note that Theorem 5.1 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 Remark 5.2 we have the following.

Corollary 5.7. Assume that {ℱ_𝑛}_𝑛≥0 is regular. Let 1≤𝑝<∞, −1/𝑝≤𝜆<0, and 𝜇=𝜆+𝛼<0. Then 𝐼𝐼_𝛼∈𝐵𝐿_𝑝,𝜆,𝐿_𝑞,𝜇,if𝑝>1,1≤𝑞≤(𝜆/𝜇)𝑝,_𝛼𝐼∈𝐵𝐿_1,𝜆,𝐿_𝑞,𝜇,if𝑝=1,1≤𝑞<(𝜆/𝜇)𝑝,_𝛼∈𝐵𝐿_1,𝜆,𝑊𝐿_𝑞,𝜇,if𝑝=1,𝑞=(𝜆/𝜇)𝑝.(5.11)

For Campanato spaces, one has the following.

Theorem 5.8. Assume that {ℱ_𝑛}_𝑛≥0 is regular. Let 1≤𝑝<∞, 𝜆≥−1/𝑝, 1≤𝑞<∞, and 𝜇=𝜆+𝛼≥0. Then ‖‖𝐼_𝛼𝑓‖‖_ℒ𝑞,𝜇≤𝐶‖𝑓‖_ℒ𝑝,𝜆,𝑓∈𝐿₁,(5.12) and 𝐼_𝛼∈𝐵ℒ_𝑝,𝜆,ℒ_𝑞,𝜇,(5.13) where 𝐼_𝛼𝑓 in (5.12) denotes the limit function described in Remark 5.3, for 𝑓=(𝐸_𝑛𝑓)_𝑛≥0, 𝑓∈𝐿₁.

Remark 5.9. If 𝜇<0, then, by Theorem 3.1 (ii), the inequalities (5.8) and (5.9) in Theorem 5.5 hold with 𝐿_𝑝,𝜆 and 𝐿_𝑞,𝜇 replaced by ℒ_𝑝,𝜆 and ℒ_𝑞,𝜇, respectively.
If 1<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 𝐿_𝑝⫋𝐿_1,𝜆 in general, the following is an improvement of the results in [21].

Corollary 5.10. Assume that {ℱ_𝑛}_𝑛≥0 is regular. Then, for 𝑓∈𝐿₁, ‖‖𝐼_𝛼𝑓‖‖_BMO≤𝐶‖𝑓‖_{𝐿1,−1/𝑝},if−1‖‖𝐼𝑝+𝛼=0,1<𝑝<∞,_𝛼𝑓‖‖_Lip𝛽≤𝐶‖𝑓‖_{𝐿1,−1/𝑝},if−1‖‖𝐼𝑝+𝛼=𝛽>0,1<𝑝<∞,_𝛼‖‖𝐼_𝛼𝑓‖‖_Lip𝛼≤𝐶‖𝑓‖_BMO,𝐼𝑓‖‖_Lip𝛾≤𝐶‖𝑓‖_Lip𝛽,if𝛽+𝛼=𝛾,𝛽>0,(5.14)_𝛼∈𝐵𝐿_1,−1/𝑝,BMO,if−1𝐼𝑝+𝛼=0,1<𝑝<∞,_𝛼∈𝐵𝐿_1,−1/𝑝,Lip_𝛽,if−1𝐼𝑝+𝛼=𝛽>0,1<𝑝<∞,_𝛼𝐼_𝛼∈𝐵BMO,Lip_𝛼,∈𝐵Lip_𝛽,Lip_𝛾,if𝛽+𝛼=𝛾,𝛽>0.(5.15)

5.2. Proof of Theorem 5.5

By Remark 5.6 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𝐼_𝛼𝑓^∗(𝜔)≤𝐶𝑓^∗(𝜔)^𝜇/𝜆‖𝑓‖^{𝐿−𝛼/𝜆}_𝑝,𝜆,(5.16)

where 𝐶 is a positive constant independent of 𝜔. To do this we prove that, for any 𝑚≥1 and any 𝐵_𝑚∈𝐴(ℱ_𝑚),||𝐼_𝛼𝑓_𝑚(𝜔)||≤𝐶𝑓^∗(𝜔)^𝜇/𝜆‖𝑓‖^{𝐿−𝛼/𝜆}_𝑝,𝜆,𝜔∈𝐵_𝑚.(5.17)

Take 𝐵_𝑘∈𝐴(ℱ_𝑘), 0≤𝑘<𝑚, such that 𝐵_𝑚⊂𝐵_𝑚−1⊂…⊂𝐵₀ and let𝐾=𝑘∶0<𝑘≤𝑚,𝐵_𝑘≠𝐵_𝑘−1=𝑘₁,𝑘₂,…,𝑘_ℓ,(5.18)

where 0=𝑘₀<𝑘₁<𝑘₂<…<𝑘_ℓ. Then, by Lemma 3.3,1+𝑅1𝑏^𝑘𝑗≤𝑏_𝑘𝑗−1≤𝑅𝑏_𝑘𝑗on𝐵_𝑚.(5.19)

Hence, we can write𝐼_𝛼𝑓_𝑚=_{0<𝑘𝑗≤𝑚}𝑏^𝛼𝑘_𝑗−1𝑑_𝑘𝑗𝑓=^ℓ_𝑗=1𝑏^𝛼𝑘_𝑗−1𝑑_𝑘𝑗𝑓on𝐵_𝑚,(5.20)

since 𝑏_𝑘=𝑏_𝑘−1 and 𝑑_𝑘𝑓=0 for 𝑘∉𝐾. Note that for 𝜔∈𝐵_𝑚,|||𝑑_𝑘𝑗𝑓(𝜔)|||==|||||||||𝑓_𝑘𝑗(1𝜔)−𝑓_𝑘𝑗−1(𝜔)|||≤|||𝑓_𝑘𝑗(𝜔)|||+|||𝑓_𝑘𝑗−1(𝜔)|||𝑃𝐵_𝑘𝑗^𝐵𝑘𝑗𝑓𝑑𝑃||||||+||||||1𝑃𝐵_𝑘𝑗−1^{𝐵𝑘𝑗−1}≤⎛⎜⎜⎝1𝑓𝑑𝑃||||||𝑃𝐵_𝑘𝑗^𝐵𝑘𝑗||𝑓||^𝑝𝑑𝑃⎞⎟⎟⎠^1/𝑝+⎛⎜⎜⎝1𝑃𝐵_𝑘𝑗−1^{𝐵𝑘𝑗−1}||𝑓||^𝑝𝑑𝑃⎞⎟⎟⎠^1/𝑝≤𝑃𝐵_𝑘𝑗^𝜆+𝑃𝐵_𝑘𝑗−1^𝜆‖𝑓‖^{𝐿𝑝,𝜆}≤1+1𝑅^𝜆𝑏^𝑘𝑗−1(𝜔)^𝜆‖𝑓‖_{𝐿𝑝,𝜆}.(5.21)

Then, for 𝜔∈𝐵_𝑚 and for 0<𝑛≤𝑚,|||||_𝑛_𝑘=0𝑏_𝑘−1(𝜔)^𝛼𝑑_𝑘𝑓(𝜔)|||||≤1+1𝑅^𝜆_{0<𝑘𝑗≤𝑛}𝑏_𝑘𝑗−1(𝜔)^𝜇‖𝑓‖_{𝐿𝑝,𝜆}≤1+(1/𝑅)^𝜆1−(1+1/𝑅)^𝜇𝑏_𝑛(𝜔)^𝜇‖𝑓‖_{𝐿𝑝,𝜆}.(5.22)

On the other hand, for 0≤𝑛<𝑚, letting 𝑗(𝑛)=min{𝑗∶𝑛<𝑘_𝑗}, we have, for 𝜔∈𝐵_𝑚,^𝑚_𝑘=𝑛+1𝑏_𝑘−1(𝜔)^𝛼𝑑_𝑘𝑓(𝜔)=^ℓ_𝑗=_𝑗(𝑛)=𝑏_𝑘𝑗−1(𝜔)^𝛼𝑑_𝑘𝑗𝑓(𝜔)_ℓ𝑗=_𝑗(𝑛)𝑏_𝑘𝑗−1(𝜔)^𝛼𝑓_𝑘𝑗(𝜔)−^ℓ_𝑗=_𝑗(𝑛)=𝑏_𝑘ℓ−1𝑏_𝑘𝑗−1(𝜔)^𝛼𝑓_𝑘𝑗−1(𝜔)(𝜔)^𝛼𝑓_𝑘ℓ(𝜔)+^ℓ−1_𝑗=_𝑗(𝑛)𝑏^𝑘_𝑗−1−𝑏_{𝑘𝑗(𝑛)−1}(𝜔)^𝛼−𝑏_𝑘𝑗(𝜔)^𝛼𝑓^𝑘_𝑗(𝜔)|||||(𝜔)^𝛼𝑓_{𝑘𝑗(𝑛)−1}(𝜔),(5.23)_𝑚𝑘_=𝑛+1𝑏_𝑘−1(𝜔)^𝛼𝑑_𝑘𝑓(𝜔)|||||≤+𝑏_𝑘ℓ−1(𝜔)^𝛼𝑓^∗(𝜔)_ℓ−1𝑗=_𝑗(𝑛)|||𝑏_𝑘𝑗−1(𝜔)^𝛼−𝑏_𝑘𝑗(𝜔)^𝛼|||𝑓_∗+𝑏_{𝑘𝑗(𝑛)−1}(𝜔)≤2𝑏_{𝑘𝑗(𝑛)−1}(𝜔)^𝛼𝑓^∗(𝜔)(𝜔)^𝛼𝑓^∗(𝜔)=2𝑏_𝑛(𝜔)^𝛼𝑓^∗(𝜔).(5.24)

Here, letΩ₁=⎧⎪⎨⎪⎩𝜔∈Ω∶𝑓^∗(𝜔)‖𝑓‖_{𝐿𝑝,𝜆}^1/𝜆≤𝑏₀(𝜔)⎫⎪⎬⎪⎭,Ω₂=Ω⧵Ω₁.(5.25)

If 𝜔∈Ω₁∩𝐵_𝑚 and𝑓^∗(𝜔)‖𝑓‖_{𝐿𝑝,𝜆}^1/𝜆≤𝑏_𝑚(𝜔),(5.26)

then, by (5.22) we have||𝐼_𝛼𝑓_𝑚(𝜔)||≲𝑏_𝑚(𝜔)^𝜇‖𝑓‖_{𝐿𝑝,𝜆}≤𝑓^∗(𝜔)‖𝑓‖_{𝐿𝑝,𝜆}^𝜇/𝜆‖𝑓‖_{𝐿𝑝,𝜆}=𝑓^∗(𝜔)^𝜇/𝜆‖𝑓‖^{𝐿−𝛼/𝜆}_𝑝,𝜆,(5.27)

since 𝜇<0 and 1−𝜇/𝜆=−𝛼/𝜆. If 𝜔∈Ω₁∩𝐵_𝑚 and𝑏_𝑚(𝜔)<𝑓^∗(𝜔)‖𝑓‖_{𝐿𝑝,𝜆}^1/𝜆,(5.28)

choosing 𝑛 such that𝑅⁻¹𝑏_𝑛(𝜔)<𝑓^∗(𝜔)‖𝑓‖_{𝐿𝑝,𝜆}^1/𝜆≤𝑏_𝑛(𝜔),(5.29)

we have by (5.22) and (5.24)||𝐼_𝛼𝑓_𝑚(𝜔)||≲≲𝑏_𝑛𝑓(𝜔)^𝜇‖𝑓‖_{𝐿𝑝,𝜆}+𝑏_𝑛(𝜔)^𝛼𝑓^∗(𝜔)_∗(𝜔)‖𝑓‖_{𝐿𝑝,𝜆}^𝜇/𝜆‖𝑓‖_{𝐿𝑝,𝜆}+𝑓^∗(𝜔)‖𝑓‖_{𝐿𝑝,𝜆}^𝛼/𝜆≲𝑓_∗𝑓^∗(𝜔)(𝜔)^𝜇/𝜆‖𝑓‖^{𝐿−𝛼/𝜆}_𝑝,𝜆.(5.30)

If 𝜔∈Ω₂∩𝐵_𝑚, then by (5.24) we have||𝐼_𝛼𝑓_𝑚(𝜔)||=|||||^𝑚_𝑘=1≤𝑓𝑏_𝑘−1(𝜔)^𝛼𝑑_𝑘𝑓(𝜔)|||||≲𝑏₀(𝜔)^𝛼𝑓^∗(𝜔)_∗(𝜔)‖𝑓‖_{𝐿𝑝,𝜆}^𝛼/𝜆𝑓^∗(𝜔)=𝑓^∗(𝜔)^𝜇/𝜆‖𝑓‖^{𝐿−𝛼/𝜆}_𝑝,𝜆.(5.31)

Therefore, we have (5.17).

Next, applying the boundedness of the maximal function (Theorem 4.1), we show𝑃1(𝐵)_𝐵𝐼_𝛼sup_𝑡>0𝑓(𝜔)^∗^𝑞𝑑𝑃^1/𝑞≲𝑃(𝐵)^𝜇‖𝑓‖_{𝐿𝑝,𝜆},if𝑝>1,(5.32)𝑡𝑃𝐵∩𝐼^𝛼𝑓^∗>𝑡𝑃(𝐵)^1/𝑞≲𝑃(𝐵)^𝜇‖𝑓‖_𝐿1,𝜆,if𝑝=1,(5.33)

for any 𝑛 and any 𝐵∈𝐴(ℱ_𝑛). To show (5.32), using (5.16) and the relations 𝑞𝜇/𝜆=𝑝 and −𝛼/𝜆=1−𝑝/𝑞, we have𝑃1(𝐵)_𝐵𝐼_𝛼𝑓(𝜔)^∗^𝑞𝑑𝑃^1/𝑞≲1𝑃(𝐵)_𝐵𝑓^∗(𝜔)^{𝑞𝜇/𝜆}𝑑𝑃^1/𝑞‖𝑓‖^{𝐿−𝛼/𝜆}_𝑝,𝜆=1𝑃(𝐵)_𝐵𝑓^∗(𝜔)^𝑝𝑑𝑃^{(1/𝑝)(𝑝/𝑞)}‖𝑓‖^{𝐿1−𝑝/𝑞}_𝑝,𝜆.(5.34)

Applying the boundedness of the maximal function, we have𝑃1(𝐵)_𝐵𝑓^∗(𝜔)^𝑝𝑑𝑃^{(1/𝑝)(𝑝/𝑞)}≤𝑃(𝐵)^𝜆‖𝑓^∗‖_{𝐿𝑝,𝜆}^𝑝/𝑞≲𝑃(𝐵)^𝜇‖𝑓‖^{𝐿𝑝/𝑞}_𝑝,𝜆.(5.35)

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 havesup_𝑡>0𝑡𝑃𝐵∩𝐼^𝛼𝑓^∗>𝑡𝑃(𝐵)^1/𝑞≲sup_𝑡>0𝑡⎛⎜⎜⎝𝑃𝐵∩𝑓^∗^𝜇/𝜆‖𝑓‖^{𝐿−𝛼/𝜆}_1,𝜆>𝑡𝑃(𝐵)⎞⎟⎟⎠^1/𝑞=sup_𝑡>0𝑡⎛⎜⎜⎝𝑃𝐵∩𝑓^∗^1/𝑞‖𝑓‖^{𝐿1−1/𝑞}_1,𝜆>𝑡𝑃(𝐵)⎞⎟⎟⎠^1/𝑞=sup_𝑡>0𝑡⎛⎜⎜⎝𝑃𝐵∩𝑓^∗^1/𝑞>𝑡𝑃(𝐵)⎞⎟⎟⎠^1/𝑞‖𝑓‖^{𝐿1−1/𝑞}_1,𝜆=sup_𝑡>0𝑡𝑃𝐵∩𝑓^∗>𝑡𝑃(𝐵)^1/𝑞‖𝑓‖^{𝐿1−1/𝑞}_1,𝜆.(5.36)

Applying the boundedness of the maximal function, we havesup_𝑡>0𝑡𝑃𝐵∩𝑓^∗>𝑡𝑃(𝐵)^1/𝑞≤𝑃(𝐵)^𝜆‖𝑓^∗‖_{𝑊𝐿1,𝜆}^1/𝑞≲𝑃(𝐵)^𝜇‖𝑓‖^𝐿1/𝑞_𝑝,𝜆.(5.37)

Then we have (5.33). The proof is complete.

5.3. Proof of Theorem 5.1

Let 𝜆=−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𝐼_𝛼𝑓^∗(𝜔)≤𝐶𝑓^∗(𝜔)^𝑝/𝑞‖𝑓‖^{𝐿1−𝑝/𝑞}_{𝑝,−1/𝑝}≤𝐶𝑓^∗(𝜔)^𝑝/𝑞‖𝑓‖^{𝐿1−𝑝/𝑞}_𝑝.(5.38)

Applying the boundedness of the maximal function on 𝐿_𝑝, we get the conclusion.

5.4. Proof of Theorem 5.8

Note that, for 𝑓∈𝐿₁, ‖𝑓‖_ℒ𝑝,𝜆=‖𝑓−𝐸₀𝑓‖_ℒ𝑝,𝜆 and 𝐼_𝛼𝑓=𝐼_𝛼(𝑓−𝐸₀𝑓). Then it is sufficient to prove (5.12) only for 𝑓∈𝐿⁰¹. 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 ‖𝐼_𝛼𝑓‖_ℒ𝑞,𝜇∼‖𝐼_𝛼𝑓‖_ℒ1,𝜇 for 𝑞≥1. So we only need to prove thatsup_𝑚≥0‖‖𝐼_𝛼𝑓_𝑚‖‖_ℒ1,𝜇≤𝐶sup_𝑚≥0‖‖𝑓_𝑚‖‖_ℒ1,𝜆,(5.39)

for all ℒ_1,𝜆-bounded martingales 𝑓=(𝑓_𝑚)_𝑚≥0.

To prove (5.39) we show that, for any 𝑛, any 𝑚>𝑛 and any 𝐵_𝑛∈𝐴(ℱ_𝑛),_𝐵𝑛||𝐼_𝛼𝑓_𝑚−𝐼_𝛼𝑓_𝑛||𝑑𝑃≲𝑃𝐵_𝑛^1+𝜇‖‖𝑓_𝑚‖‖_ℒ1,𝜆.(5.40)

Note that𝐼_𝛼𝑓_𝑚−𝐼_𝛼𝑓_𝑛=^𝑚_𝑘=𝑛+1𝑏^𝛼𝑘−1𝑑_𝑘𝑓=^𝑚−1_𝑘=𝑛+1𝑏^𝛼𝑘−1−𝑏^𝛼𝑘𝑓_𝑘−𝑓_𝑛+𝑏^𝛼𝑚−1𝑓_𝑚−𝑓_𝑛.(5.41)

Since 𝑏_𝑚 is nonincreasing with respect to 𝑚, we have_𝐵𝑛𝑏^𝛼𝑚−1||𝑓_𝑚−𝑓_𝑛||𝑑𝑃≤𝑃𝐵_𝑛^𝛼_𝐵𝑛||𝑓_𝑚−𝑓_𝑛||𝑑𝑃≲𝑃𝐵_𝑛^1+𝜇‖‖𝑓_𝑚‖‖_ℒ1,𝜆.(5.42)

Therefore, to prove (5.40), we only need to show that^𝑚_𝑘=𝑛+1_𝐵𝑛𝑏^𝛼𝑘−1−𝑏^𝛼𝑘||𝑓_𝑘−𝑓_𝑛||𝑑𝑃≤𝑃𝐵_𝑛^1+𝜇‖‖𝑓_𝑚‖‖_ℒ1,𝜆∀𝑚>𝑛.(5.43)

Using the inequality |𝑓_𝑘−𝑓_𝑛|≤𝐸_𝑘|𝑓_𝑚−𝑓_𝑛| for 𝑛<𝑘≤𝑚, we have_𝐵𝑛𝑏^𝛼𝑘−1−𝑏^𝛼𝑘||𝑓_𝑘−𝑓_𝑛||𝑑𝑃≤_𝐵𝑛𝑏^𝛼𝑘−1−𝑏^𝛼𝑘𝐸_𝑘||𝑓_𝑚−𝑓_𝑛||𝑑𝑃=_𝐵𝑛𝑏^𝛼𝑘−1−𝑏^𝛼𝑘||𝑓_𝑚−𝑓_𝑛||𝑑𝑃.(5.44)

Therefore, we have^𝑚_𝑘=𝑛+1_𝐵𝑛𝑏^𝛼𝑘−1−𝑏^𝛼𝑘||𝑓_𝑘−𝑓_𝑛||𝑑𝑃≤^𝑚_𝑘_=𝑛+1_𝐵𝑛𝑏^𝛼𝑘−1=−𝑏^𝛼𝑘||𝑓_𝑚−𝑓_𝑛||𝑑𝑃_𝐵𝑛≤𝑃𝐵_𝑛𝑏^𝛼𝑛−𝑏^𝛼𝑚||𝑓_𝑚−𝑓_𝑛||𝑑𝑃≤𝑃𝐵_𝑛^𝛼_𝐵𝑛||𝑓_𝑚−𝑓_𝑛||𝑑𝑃^1+𝜇‖‖𝑓_𝑚‖‖_ℒ1,𝜆.(5.45)

We have obtained (5.43). The proof of Theorem 5.8 is completed.

Acknowledgments

The authors wish to express their deep thanks to the referees for their very careful reading and also their many valuable and suggested remarks, which led them to simplify the proofs of the main results and improve the presentation of this article. The first author was supported by Grant-in-Aid for Scientific Research (C), no. 20540167 and no. 24540159, Japan Society for the Promotion of Science. The second author was supported by Grant-in-Aid for Scientific Research (C), no. 24540171, Japan Society for the Promotion of Science.