Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 673929 |

Eiichi Nakai, Gaku Sadasue, "Martingale Morrey-Campanato Spaces and Fractional Integrals", Journal of Function Spaces, vol. 2012, Article ID 673929, 29 pages, 2012.

Martingale Morrey-Campanato Spaces and Fractional Integrals

Academic Editor: Dachun Yang
Received03 Mar 2012
Revised23 Apr 2012
Accepted23 Apr 2012
Published09 Jul 2012


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 𝐿𝑝 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 (Ξ©,β„±,𝑃) such that ⋃ℱ=𝜎(𝑛ℱ𝑛), where {ℱ𝑛}𝑛β‰₯0 is a nondecreasing sequence of sub-𝜎-algebras of β„±. Following Weisz [5], we call {ℱ𝑛}𝑛β‰₯0 a stochastic basis. For the sake of simplicity, let β„±βˆ’1=β„±0. We suppose that every 𝜎-algebra ℱ𝑛 is generated by countable atoms, where π΅βˆˆβ„±π‘› is called an atom (more precisely a (ℱ𝑛,𝑃)-atom), if any π΄βŠ‚π΅ with π΄βˆˆβ„±π‘› satisfies 𝑃(𝐴)=𝑃(𝐡) 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: let π‘βˆˆ[1,∞) and πœ†βˆˆ(βˆ’βˆž,∞). For π‘“βˆˆπΏ1, let‖𝑓‖𝐿𝑝,πœ†=sup𝑛β‰₯0supξ€·β„±π΅βˆˆπ΄π‘›ξ€Έ1𝑃(𝐡)πœ†ξ‚΅1ξ€œπ‘ƒ(𝐡)𝐡||𝑓||𝑝𝑑𝑃1/𝑝,‖𝑓‖ℒ𝑝,πœ†=sup𝑛β‰₯0supξ€·β„±π΅βˆˆπ΄π‘›ξ€Έ1𝑃(𝐡)πœ†ξ‚΅1π‘ƒξ€œ(𝐡)𝐡||π‘“βˆ’πΈπ‘›π‘“||𝑝𝑑𝑃1/𝑝,(1.1) and let𝐿𝑝,πœ†=ξ‚†π‘“βˆˆπΏ0π‘βˆΆβ€–π‘“β€–πΏπ‘,πœ†ξ‚‡<∞,ℒ𝑝,πœ†=ξ‚†π‘“βˆˆπΏ0π‘βˆΆβ€–π‘“β€–β„’π‘,πœ†ξ‚‡<∞,(1.2) where 𝐿0𝑝 is the set of all π‘“βˆˆπΏπ‘ 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-πΏπ‘ž 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 martingale 𝑓=(𝑓𝑛)𝑛β‰₯0 relative to {ℱ𝑛}𝑛β‰₯0, denote its martingale difference by 𝑑𝑛𝑓=π‘“π‘›βˆ’π‘“π‘›βˆ’1 (𝑛β‰₯0, with convention 𝑑0𝑓=0). For 𝛼>0, we define the fractional integral 𝐼𝛼𝑓=((𝐼𝛼𝑓)𝑛)𝑛β‰₯0 of 𝑓 by𝐼𝛼𝑓𝑛=π‘›ξ“π‘˜=0π‘π›Όπ‘˜βˆ’1π‘‘π‘˜π‘“,(1.3) where π‘π‘˜ is an β„±π‘˜-measurable function such thatπ‘π‘˜ξ€·β„±(πœ”)=𝑃(𝐡)fora.e.πœ”βˆˆπ΅withπ΅βˆˆπ΄π‘˜ξ€Έ.(1.4) Then ((𝐼𝛼𝑓)𝑛)𝑛β‰₯0 is a martingale for any martingale 𝑓=(𝑓𝑛)𝑛β‰₯0, since each π‘π‘˜ 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 πΏπ‘ž, if 1<𝑝<π‘ž<∞ and βˆ’1/𝑝+𝛼=βˆ’1/π‘ž. That is, if a martingale 𝑓=(𝑓𝑛)𝑛β‰₯0 is 𝐿𝑝-bounded, then ((𝐼𝛼𝑓)𝑛)𝑛β‰₯0 is πΏπ‘ž-bounded and the following inequality holds:sup𝑛β‰₯0β€–β€–ξ€·πΌπ›Όπ‘“ξ€Έπ‘›β€–β€–πΏπ‘žβ‰€πΆsup𝑛β‰₯0‖‖𝑓𝑛‖‖𝐿𝑝,(1.5) where 𝐢 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 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 𝐢 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 𝐢𝑝, is dependent on the subscripts. If 𝑓≀𝐢𝑔, we then write 𝑓≲𝑔 or 𝑔≳𝑓 and if 𝑓≲𝑔≲𝑓, we then write π‘“βˆΌπ‘”.

2. Notation, Definitions, and Remarks

Recall that (Ξ©,β„±,𝑃) is a probability space, and {ℱ𝑛}𝑛β‰₯0 a nondecreasing sequence of sub-𝜎-algebras of β„± such that ⋃ℱ=𝜎(𝑛ℱ𝑛). For the sake of simplicity, let β„±βˆ’1=β„±0. 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 martingale 𝑓=(𝑓𝑛)𝑛β‰₯0 relative to {ℱ𝑛}𝑛β‰₯0, the maximal function π‘“βˆ— of 𝑓 is defined byπ‘“βˆ—π‘›=sup0β‰€π‘šβ‰€π‘›||π‘“π‘š||,π‘“βˆ—=sup𝑛β‰₯0||𝑓𝑛||.(2.1)

It is known that if π‘βˆˆ(1,∞), then any 𝐿𝑝-bounded martingale converges in 𝐿𝑝. Moreover, if π‘“βˆˆπΏπ‘, π‘βˆˆ[1,∞), then (𝑓𝑛)𝑛β‰₯0 with 𝑓𝑛=𝐸𝑛𝑓 is an 𝐿𝑝-bounded martingale and converges to 𝑓 in 𝐿𝑝 (see, e.g., [26]). For this reason a function π‘“βˆˆπΏ1 and the corresponding martingale (𝑓𝑛)𝑛β‰₯0 with 𝑓𝑛=𝐸𝑛𝑓 will be denoted by the same symbol 𝑓. Note also that ‖𝑓‖𝐿𝑝=sup𝑛β‰₯0‖𝐸𝑛𝑓‖𝐿𝑝. In this caseπ‘“βˆ—π‘›=sup0β‰€π‘šβ‰€π‘›||πΈπ‘šπ‘“||,π‘“βˆ—=sup𝑛β‰₯0||𝐸𝑛𝑓||forπ‘“βˆˆπΏ1.(2.2)

Let β„³ be the set of all martingales such that 𝑓0=0. For π‘βˆˆ[1,∞], let 𝐿0𝑝 be the set of all π‘“βˆˆπΏπ‘ such that 𝐸0𝑓=0. For any π‘“βˆˆπΏ0𝑝, let 𝑓𝑛=𝐸𝑛𝑓. Then (𝑓𝑛)𝑛β‰₯0 is an 𝐿𝑝-bounded martingale in β„³. For this reason we can regard 𝐿0𝑝 as a subset of β„³.

In Section 1 we have introduced Morrey spaces 𝐿𝑝,πœ† and Campanato spaces ℒ𝑝,πœ† as the following.

Definition 2.1. Let π‘βˆˆ[1,∞) and πœ†βˆˆ(βˆ’βˆž,∞). For π‘“βˆˆπΏ1, let ‖𝑓‖𝐿𝑝,πœ†=sup𝑛β‰₯0sup𝐡∈𝐴(ℱ𝑛)1𝑃(𝐡)πœ†ξ‚΅1ξ€œπ‘ƒ(𝐡)𝐡||𝑓||𝑝𝑑𝑃1/𝑝,‖𝑓‖ℒ𝑝,πœ†=sup𝑛β‰₯0supξ€·β„±π΅βˆˆπ΄π‘›ξ€Έ1𝑃(𝐡)πœ†ξ‚΅1ξ€œπ‘ƒ(𝐡)𝐡||π‘“βˆ’πΈπ‘›π‘“||𝑝𝑑𝑃1/𝑝,(2.3) and define 𝐿𝑝,πœ†=ξ‚†π‘“βˆˆπΏ0π‘βˆΆβ€–π‘“β€–πΏπ‘,πœ†ξ‚‡<∞,ℒ𝑝,πœ†=ξ‚†π‘“βˆˆπΏ0π‘βˆΆβ€–π‘“β€–β„’π‘,πœ†ξ‚‡<∞.(2.4)
Then functionals ‖𝑓‖𝐿𝑝,πœ† and ‖𝑓‖ℒ𝑝,πœ† are norms on 𝐿𝑝,πœ† and ℒ𝑝,πœ†, respectively. Note that 𝐿𝑝,πœ† and ℒ𝑝,πœ† are not always trivial set {0} even if πœ†>0 and πœ†>1, respectively. This property is different from classical Morrey-Campanato spaces on ℝ𝑛.
The martingale 𝑓=(𝑓𝑛)𝑛β‰₯0 is said to be 𝐿𝑝,πœ†-bounded if π‘“π‘›βˆˆπΏπ‘,πœ† (𝑛β‰₯0) and sup𝑛β‰₯0‖𝑓𝑛‖𝐿𝑝,πœ†<∞. Similarly, the martingale 𝑓=(𝑓𝑛)𝑛β‰₯0 is said to be ℒ𝑝,πœ†-bounded if π‘“π‘›βˆˆβ„’π‘,πœ† (𝑛β‰₯0) and sup𝑛β‰₯0‖𝑓𝑛‖ℒ𝑝,πœ†<∞.

Proposition 2.2. Let 1≀𝑝<∞. Let π‘“βˆˆπΏ1 and (𝑓𝑛)𝑛β‰₯0 be its corresponding martingale with 𝑓𝑛=𝐸𝑛𝑓 (𝑛β‰₯0).
(i)Assume that πœ†βˆˆ(βˆ’βˆž,0]. If π‘“βˆˆπΏπ‘,πœ†, then (𝑓𝑛)𝑛β‰₯0 is 𝐿𝑝,πœ†-bounded and ‖𝑓‖𝐿𝑝,πœ†β‰₯sup𝑛β‰₯0‖‖𝑓𝑛‖‖𝐿𝑝,πœ†.(2.5) Conversely, if (𝑓𝑛)𝑛β‰₯0 is 𝐿𝑝,πœ†-bounded, then π‘“βˆˆπΏπ‘,πœ† and ‖𝑓‖𝐿𝑝,πœ†β‰€sup𝑛β‰₯0‖‖𝑓𝑛‖‖𝐿𝑝,πœ†.(2.6)(ii)Assume that πœ†βˆˆ(βˆ’βˆž,∞). If π‘“βˆˆβ„’π‘,πœ†, then (𝑓𝑛)𝑛β‰₯0 is ℒ𝑝,πœ†-bounded and ‖𝑓‖ℒ𝑝,πœ†β‰₯sup𝑛β‰₯0‖‖𝑓𝑛‖‖ℒ𝑝,πœ†.(2.7) Conversely, if (𝑓𝑛)𝑛β‰₯0 is ℒ𝑝,πœ†-bounded, then π‘“βˆˆβ„’π‘,πœ† and ‖𝑓‖ℒ𝑝,πœ†β‰€sup𝑛β‰₯0‖‖𝑓𝑛‖‖ℒ𝑝,πœ†.(2.8)

Proof. (i) Let π‘“βˆˆπΏπ‘,πœ† and 𝑛β‰₯0. Fix any 𝐡∈𝐴(β„±π‘˜). If π‘˜β‰€π‘›, then ξ‚΅ξ€œπ΅||𝐸𝑛𝑓||𝑝𝑑𝑃1/π‘β‰€ξ‚΅ξ€œπ΅πΈπ‘›ξ€Ί||𝑓||𝑝𝑑𝑃1/𝑝=ξ‚΅ξ€œπ΅||𝑓||𝑝𝑑𝑃1/𝑝≀𝑃(𝐡)πœ†+1/𝑝‖𝑓‖𝐿𝑝,πœ†.(2.9) If 𝑛<π‘˜, then taking π΅π‘›βˆˆπ΄(ℱ𝑛) such that π΅βŠ‚π΅π‘›, we have 𝐸𝑛1𝑓=π‘ƒξ€·π΅π‘›ξ€Έξ€œπ΅π‘›π‘“π‘‘π‘ƒon𝐡𝑛.(2.10) Hence ξ‚΅1π‘ƒξ€œ(𝐡)𝐡||𝐸𝑛𝑓||𝑝𝑑𝑃1/𝑝=1π‘ƒξ€·π΅π‘›ξ€Έξ€œπ΅π‘›||𝐸𝑛𝑓||𝑝ξƒͺ𝑑𝑃1/𝑝≀1π‘ƒξ€·π΅π‘›ξ€Έξ€œπ΅π‘›πΈπ‘›ξ€Ί||𝑓||𝑝ξƒͺ𝑑𝑃1/𝑝=1π‘ƒξ€·π΅π‘›ξ€Έξ€œπ΅π‘›||𝑓||𝑝ξƒͺ𝑑𝑃1/π‘ξ€·π΅β‰€π‘ƒπ‘›ξ€Έπœ†β€–π‘“β€–πΏπ‘,πœ†β‰€π‘ƒ(𝐡)πœ†β€–π‘“β€–πΏπ‘,πœ†.(2.11) Therefore, 𝑓𝑛=πΈπ‘›π‘“βˆˆπΏπ‘,πœ† and ‖𝑓𝑛‖𝐿𝑝,πœ†β‰€β€–π‘“β€–πΏπ‘,πœ† for all 𝑛β‰₯0. This shows that (𝑓𝑛)𝑛β‰₯0 is a 𝐿𝑝,πœ†-bounded martingale and sup𝑛β‰₯0‖‖𝑓𝑛‖‖𝐿𝑝,πœ†β‰€β€–π‘“β€–πΏπ‘,πœ†.(2.12) Conversely, from the inequality ξ€œπ΅||𝑓||𝑝𝑑𝑃≀liminfπ‘›β†’βˆžξ€œπ΅||𝐸𝑛𝑓||π‘ξšπ‘‘π‘ƒβˆ€π΅βˆˆπ‘šπ΄ξ€·β„±π‘šξ€Έ,(2.13) it follows that ‖𝑓‖𝐿𝑝,πœ†β‰€sup𝑛β‰₯0‖‖𝐸𝑛𝑓‖‖𝐿𝑝,πœ†.(2.14)(ii) Let π‘“βˆˆβ„’π‘,πœ† and 𝑛β‰₯0. Fix any 𝐡∈𝐴(β„±π‘˜). If π‘˜β‰€π‘›, thenξ‚΅ξ€œπ΅||πΈπ‘›π‘“βˆ’πΈπ‘˜ξ€ΊπΈπ‘›π‘“ξ€»||𝑝𝑑𝑃1/𝑝=ξ‚΅ξ€œπ΅||πΈπ‘›π‘“βˆ’πΈπ‘›ξ€ΊπΈπ‘˜π‘“ξ€»||𝑝𝑑𝑃1/π‘β‰€ξ‚΅ξ€œπ΅πΈπ‘›ξ€Ί||π‘“βˆ’πΈπ‘˜π‘“||𝑝𝑑𝑃1/𝑝=ξ‚΅ξ€œπ΅||π‘“βˆ’πΈπ‘˜π‘“||𝑝𝑑𝑃1/𝑝≀𝑃(𝐡)πœ†+1/𝑝‖𝑓‖ℒ𝑝,πœ†.(2.15) If 𝑛<π‘˜, then πΈπ‘˜[𝐸𝑛𝑓]=𝐸𝑛𝑓. Hence ξ‚΅1ξ€œπ‘ƒ(𝐡)𝐡||πΈπ‘›π‘“βˆ’πΈπ‘˜ξ€ΊπΈπ‘›π‘“ξ€»||𝑝𝑑𝑃1/𝑝=0.(2.16) Therefore, 𝑓𝑛=πΈπ‘›π‘“βˆˆβ„’π‘,πœ† and ‖𝑓𝑛‖ℒ𝑝,πœ†β‰€β€–π‘“β€–β„’π‘,πœ† for all 𝑛β‰₯0. This shows that (𝑓𝑛)𝑛β‰₯0 is a ℒ𝑝,πœ†-bounded martingale and sup𝑛β‰₯0‖‖𝑓𝑛‖‖ℒ𝑝,πœ†β‰€β€–π‘“β€–β„’π‘,πœ†.(2.17) Conversely, from the inequality ξ€œπ΅||π‘“βˆ’π‘“π‘˜||𝑝𝑑𝑃≀liminfπ‘›β†’βˆžξ€œπ΅||πΈπ‘›ξ€Ίπ‘“βˆ’π‘“π‘˜ξ€»||π‘ξ€·β„±π‘‘π‘ƒβˆ€π΅βˆˆπ΄π‘˜ξ€Έ,π‘˜=1,2,…,(2.18) it follows that ‖𝑓‖ℒ𝑝,πœ†β‰€sup𝑛β‰₯0‖‖𝐸𝑛𝑓‖‖ℒ𝑝,πœ†.(2.19)

Remark 2.3. In general, for π‘“βˆˆπΏπ‘,πœ† (resp., ℒ𝑝,πœ†), 𝐸𝑛𝑓 does not always converge to 𝑓 in 𝐿𝑝,πœ† (resp., ℒ𝑝,πœ†). See Remark 3.7.

Remark 2.4. If β„±0={βˆ…,Ξ©}, then 𝐿𝑝,πœ†βŠ‚πΏπ‘ and ‖𝑓‖𝐿𝑝≀‖𝑓‖𝐿𝑝,πœ† for π‘“βˆˆπΏπ‘,πœ†. Therefore, if (𝑓𝑛)𝑛β‰₯0 is an 𝐿𝑝,πœ†-bounded martingale, then it is an 𝐿𝑝-bounded martingale. If 1<𝑝<∞, from the known result it follows that there exists π‘“βˆˆπΏ0𝑝 such that 𝐸𝑛𝑓=𝑓𝑛, 𝑛β‰₯0, and (𝑓𝑛)𝑛β‰₯0 converges to 𝑓 in 𝐿𝑝 and a.e. Moreover, we can deduce that π‘“βˆˆπΏπ‘,πœ†, since ξ‚΅ξ€œπ΅||𝑓𝑛||(πœ”)𝑝𝑑𝑃1/𝑝≀𝑃(𝐡)πœ†+1/𝑝‖‖𝑓𝑛‖‖𝐿𝑝,πœ†β‰€π‘ƒ(𝐡)πœ†+1/𝑝sup𝑛β‰₯0‖‖𝑓𝑛‖‖𝐿𝑝,πœ†ξš,π΅βˆˆπ‘šπ΄ξ€·β„±π‘šξ€Έ.(2.20) The stochastic basis {ℱ𝑛}𝑛β‰₯0 is said to be regular, if there exists a constant 𝑅β‰₯2 such that π‘“π‘›β‰€π‘…π‘“π‘›βˆ’1(2.21) holds for all nonnegative martingales (𝑓𝑛)𝑛β‰₯0.

Remark 2.5. In general, 𝐿𝑝,πœ†βŠ‚β„’π‘,πœ† with ‖𝑓‖ℒ𝑝,πœ†β‰€2‖𝑓‖𝐿𝑝,πœ†. Actually, for any 𝐡∈𝐴(ℱ𝑛), ξ‚΅1ξ€œπ‘ƒ(𝐡)𝐡||π‘“βˆ’πΈπ‘›π‘“||𝑝𝑑𝑃1/𝑝≀1ξ€œπ‘ƒ(𝐡)𝐡||𝑓||𝑝𝑑𝑃1/𝑝+ξ‚΅1ξ€œπ‘ƒ(𝐡)𝐡||𝐸𝑛𝑓||𝑝𝑑𝑃1/𝑝1≀2ξ€œπ‘ƒ(𝐡)𝐡||𝑓||𝑝𝑑𝑃1/𝑝.(2.22) Moreover, if {ℱ𝑛}𝑛β‰₯0 is regular and πœ†<0, then we can prove that 𝐿𝑝,πœ†=ℒ𝑝,πœ† with equivalent norms (Theorem 3.1 and Remark 3.2).

Remark 2.6. By definition, if 0β‰₯πœ†β‰₯πœ†ξ…ž, then we have that 𝐿0βˆžβŠ‚πΏπ‘,πœ†βŠ‚πΏπ‘,πœ†β€² with ‖𝑓‖𝐿′𝑝,πœ†β‰€β€–π‘“β€–πΏπ‘,πœ†β‰€β€–π‘“β€–πΏβˆž and 𝐿0βˆžβŠ‚β„’π‘,πœ†βŠ‚β„’π‘,πœ†β€² with ‖𝑓‖ℒ′𝑝,πœ†β‰€β€–π‘“β€–β„’π‘,πœ†β‰€2β€–π‘“β€–πΏβˆž. If πœ†β‰€βˆ’1/𝑝, then 𝐿0π‘βŠ‚πΏπ‘,πœ†βŠ‚β„’π‘,πœ† with ‖𝑓‖ℒ𝑝,πœ†/2≀‖𝑓‖𝐿𝑝,πœ†β‰€β€–π‘“β€–πΏπ‘.

Remark 2.7. By definition and Remark 2.5, if β„±0={βˆ…,Ξ©}, then 𝐿𝑝,πœ†βŠ‚β„’π‘,πœ†βŠ‚πΏ0𝑝 with ‖𝑓‖𝐿𝑝≀‖𝑓‖ℒ𝑝,πœ†β‰€2‖𝑓‖𝐿𝑝,πœ†.

Definition 2.8. Let BMO=β„’1,0 and Lip𝛼=β„’1,𝛼 if 𝛼>0.
Our definitions of BMO and Lip𝛼 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 π‘βˆˆ[1,∞) and πœ†βˆˆ(βˆ’βˆž,∞). For π‘“βˆˆπΏ1, let ‖𝑓‖𝐿𝑝,πœ†,β„±=sup𝑛β‰₯0supπ΅βˆˆβ„±π‘›1𝑃(𝐡)πœ†ξ‚΅1ξ€œπ‘ƒ(𝐡)𝐡||𝑓||𝑝𝑑𝑃1/𝑝,‖𝑓‖ℒ𝑝,πœ†,β„±=sup𝑛β‰₯0supπ΅βˆˆβ„±π‘›1𝑃(𝐡)πœ†ξ‚΅1ξ€œπ‘ƒ(𝐡)𝐡||π‘“βˆ’πΈπ‘›π‘“||𝑝𝑑𝑃1/𝑝,(2.23) and define𝐿𝑝,πœ†,β„±=ξ‚†π‘“βˆˆπΏ0π‘βˆΆβ€–π‘“β€–πΏπ‘,πœ†,ℱ<∞,ℒ𝑝,πœ†,β„±=ξ‚†π‘“βˆˆπΏ0π‘βˆΆβ€–π‘“β€–β„’π‘,πœ†,ℱ<∞.(2.24)
Note that the spaces 𝐿𝑝,πœ†,β„± and ℒ𝑝,πœ†,β„± 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 relations 𝐿𝑝,πœ†,β„±βŠ‚πΏπ‘,πœ† with ‖𝑓‖𝐿𝑝,πœ†β‰€β€–π‘“β€–πΏπ‘,πœ†,β„± and ℒ𝑝,πœ†,β„±βŠ‚β„’π‘,πœ† with ‖𝑓‖ℒ𝑝,πœ†β‰€β€–π‘“β€–β„’π‘,πœ†,β„±. If πœ†β‰₯0, then we can prove that 𝐿𝑝,πœ†,β„±=𝐿𝑝,πœ† and ℒ𝑝,πœ†,β„±=ℒ𝑝,πœ† with the same norms, respectively (see Proposition 3.8). If βˆ’1/𝑝<πœ†<0, then 𝐿𝑝,πœ†,ℱ⫋𝐿𝑝,πœ† and ℒ𝑝,πœ†,ℱ⫋ℒ𝑝,πœ† in general (see Proposition 3.9).

Remark 2.11. It is known that, if {ℱ𝑛}𝑛β‰₯0 is regular and πœ†β‰₯0, then β„’1,πœ†,β„±=ℒ𝑝,πœ†,β„± with ‖𝑓‖ℒ1,πœ†,ℱ≀‖𝑓‖ℒ𝑝,πœ†,ℱ≀𝐢𝑝‖𝑓‖ℒ1,πœ†,β„± for each π‘βˆˆ[1,∞) (see, e.g., [8]).
We also define weak Morrey spaces.

Definition 2.12. For π‘βˆˆ[1,∞) and πœ†βˆˆ(βˆ’βˆž,∞), let β€–π‘“β€–π‘ŠπΏπ‘,πœ†=sup𝑛β‰₯0sup𝐡∈𝐴(ℱ𝑛)1𝑃(𝐡)πœ†sup𝑑>0𝑑𝑃||𝑓||𝐡∩>𝑑𝑃ξƒͺ(𝐡)1/𝑝,(2.25) for measurable functions 𝑓, and define π‘ŠπΏπ‘,πœ†=ξ‚†π‘“βˆˆπΏ01βˆΆβ€–π‘“β€–π‘ŠπΏπ‘,πœ†ξ‚‡<∞.(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 {ℱ𝑛}𝑛β‰₯0 be regular, β„±0={βˆ…,Ξ©} and (Ξ©,β„±,𝑃) be nonatomic. Let π‘βˆˆ[1,∞). (i)If πœ†β‰€βˆ’1/𝑝, then 𝐿𝑝,πœ†=ℒ𝑝,πœ†=𝐿0𝑝 and 12‖𝑓‖ℒ𝑝,πœ†β‰€β€–π‘“β€–πΏπ‘,πœ†=‖𝑓‖𝐿𝑝≀‖𝑓‖ℒ𝑝,πœ†.(3.1)(ii)If βˆ’1/𝑝<πœ†<0, then 𝐿0βˆžβ«‹πΏπ‘,πœ†=ℒ𝑝,πœ†β«‹πΏ0𝑝 and ‖𝑓‖𝐿𝑝≀‖𝑓‖𝐿𝑝,πœ†β‰€β€–π‘“β€–πΏβˆž,12‖𝑓‖ℒ𝑝,πœ†β‰€β€–π‘“β€–πΏπ‘,πœ†β‰€πΆβ€–π‘“β€–β„’π‘,πœ†.(3.2)(iii)If πœ†=0, then 𝐿0∞=𝐿𝑝,0⫋ℒ𝑝,0=BMO and ‖𝑓‖𝐿𝑝,0=β€–π‘“β€–πΏβˆž,‖𝑓‖BMO≀‖𝑓‖ℒ𝑝,0≀𝐢𝑝‖𝑓‖𝐡𝑀𝑂.(3.3)(iv)If πœ†>0, then {0}=𝐿𝑝,πœ†β«‹β„’π‘,πœ†=Lipπœ† and ‖𝑓‖Lipπœ†β‰€β€–π‘“β€–β„’π‘,πœ†β‰€πΆπ‘β€–π‘“β€–Lipπœ†.(3.4)

Remark 3.2. We can prove (i) without the assumption that {ℱ𝑛}𝑛β‰₯0 is regular or that (Ξ©,β„±,𝑃) is nonatomic. In (ii), we can prove that 𝐿𝑝,πœ†=ℒ𝑝,πœ† and (1/2)‖𝑓‖ℒ𝑝,πœ†β‰€β€–π‘“β€–πΏπ‘,πœ†β‰€πΆβ€–π‘“β€–β„’π‘,πœ† without the assumption that β„±0={βˆ…,Ξ©} or that (Ξ©,β„±,𝑃) is nonatomic. To show 𝐿0βˆžβ«‹β„’π‘,πœ† in (ii) and (iii), we can replace the condition that (Ξ©,β„±,𝑃) is nonatomic by a weaker condition as in Proposition 3.6(ii), which follows from the condition that (Ξ©,β„±,𝑃) is nonatomic. In (iv), we need the condition that (Ξ©,β„±,𝑃) is nonatomic to show 𝐿𝑝,πœ†={0}.

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

Lemma 3.3. Let {ℱ𝑛}𝑛β‰₯0 be regular. Then every sequence 𝐡0βŠƒπ΅1βŠƒβ‹―βŠƒπ΅π‘›βŠƒβ‹―,π΅π‘›ξ€·β„±βˆˆπ΄π‘›ξ€Έ,(3.5) has the following property: for each 𝑛β‰₯1, 𝐡𝑛=π΅π‘›βˆ’1ξ‚€1or1+π‘…ξ‚π‘ƒξ€·π΅π‘›ξ€Έξ€·π΅β‰€π‘ƒπ‘›βˆ’1𝐡≀𝑅𝑃𝑛,(3.6) where 𝑅 is the constant in (2.21).

Remark 3.4. Since π΅π‘›βˆˆπ΄(ℱ𝑛) is an (ℱ𝑛,𝑃)-atom, we always interpret π΅π‘›βˆ’1βŠƒπ΅π‘› as the inclusion modulo null sets, that is, π΅π‘›βˆ’1βŠƒπ΅π‘› means 𝑃(π΅π‘›β§΅π΅π‘›βˆ’1)=0. Therefore, 𝐡𝑛=π΅π‘›βˆ’1 means 𝑃(π΅π‘›β§΅π΅π‘›βˆ’1)=𝑃(π΅π‘›βˆ’1⧡𝐡𝑛)=0.

Remark 3.5. By the lemma we see that there exists π‘š such that π΅π‘š=𝐡𝑛 for all 𝑛β‰₯π‘š, if and only if limπ‘›β†’βˆžπ‘ƒ(𝐡𝑛)>0.

Proof of Lemma 3.3. Let ξ‚π΅π‘›βˆ’1={πΈπ‘›βˆ’1[πœ’π΅π‘›]β‰₯1/𝑅}. Then ξ‚π΅π‘›βˆ’1βˆˆβ„±π‘›βˆ’1. By the regularity we have πœ’π΅π‘›β‰€π‘…πΈπ‘›βˆ’1[πœ’π΅π‘›]. This shows ξ‚π΅π‘›βˆ’1βŠƒπ΅π‘›. In this case, ξ‚π΅π‘›βˆ’1βŠƒπ΅π‘›βˆ’1βŠƒπ΅π‘›, since π΅π‘›βˆ’1∈𝐴(β„±π‘›βˆ’1). From the definition of ξ‚π΅π‘›βˆ’1 it follows that πœ’ξ‚π΅π‘›βˆ’1β‰€π‘…πΈπ‘›βˆ’1[πœ’π΅π‘›]. Then we have π‘ƒξ€·π΅π‘›βˆ’1ξ€Έξ‚€ξ‚π΅β‰€π‘ƒπ‘›βˆ’1ξ‚ξ‚ƒπœ’ξ‚π΅=πΈπ‘›βˆ’1ξ‚„ξ€Ίβ‰€πΈπ‘…πΈπ‘›βˆ’1ξ€Ίπœ’π΅π‘›ξ€Ίπœ’ξ€»ξ€»=𝑅𝐸𝐡𝑛𝐡=𝑅𝑃𝑛.(3.7) Next we show π΅π‘›βˆ’1=𝐡𝑛 or (1+1/𝑅)𝑃(𝐡𝑛)≀𝑃(π΅π‘›βˆ’1). Suppose that π‘ƒξ€·π΅π‘›βˆ’1ξ€Έ<ξ‚€11+𝑅𝑃𝐡𝑛.(3.8) Then π‘ƒξ€·π΅π‘›βˆ’1⧡𝐡𝑛𝐡=π‘ƒπ‘›βˆ’1ξ€Έξ€·π΅βˆ’π‘ƒπ‘›ξ€Έ<π‘ƒξ€·π΅π‘›ξ€Έπ‘…β‰€π‘ƒξ€·π΅π‘›βˆ’1𝑅.(3.9) Therefore, πΈπ‘›βˆ’1ξ€Ίπœ’π΅π‘›βˆ’1⧡𝐡𝑛=π‘ƒξ€·π΅π‘›βˆ’1β§΅π΅π‘›ξ€Έπ‘ƒξ€·π΅π‘›βˆ’1ξ€Έπœ’π΅π‘›βˆ’1<1π‘…πœ’π΅π‘›βˆ’1.(3.10) From the regularity and the inequality above it follows that πœ’π΅π‘›βˆ’1β§΅π΅π‘›β‰€π‘…πΈπ‘›βˆ’1ξ€Ίπœ’π΅π‘›βˆ’1⧡𝐡𝑛<πœ’π΅π‘›βˆ’1.(3.11) This means that π΅π‘›βˆ’1=𝐡𝑛.

Proposition 3.6. Let {ℱ𝑛}𝑛β‰₯0 be regular, 1≀𝑝<∞ and πœ†>βˆ’1/𝑝.
(i)For a sequence 𝐡0βŠƒπ΅1βŠƒβ‹―βŠƒπ΅π‘˜βŠƒβ‹―, π΅π‘˜βˆˆπ΄(β„±π‘˜), let 𝑓0=0 and 𝑓𝑛=π‘›ξ“π‘˜=1π‘ƒξ€·π΅π‘˜ξ€Έπœ†ξƒ©π‘ƒξ€·π΅π‘˜βˆ’1ξ€Έπ‘ƒξ€·π΅π‘˜ξ€Έπœ’π΅π‘˜βˆ’πœ’π΅π‘˜βˆ’1ξƒͺ,𝑛β‰₯1.(3.12)Then 𝑓=(𝑓𝑛)𝑛β‰₯0 is a martingale in β„³ and in ℒ𝑝,πœ†.(ii)Let 0β‰₯πœ†>πœ†β€²>βˆ’1/𝑝. If there exists a sequence 𝐡0βŠƒπ΅1βŠƒβ‹―βŠƒπ΅π‘˜βŠƒβ‹―, π΅π‘˜βˆˆπ΄(β„±π‘˜) and limπ‘˜β†’βˆžπ‘ƒ(π΅π‘˜)=0, then 𝐿0βˆžβ«‹β„’π‘,πœ†β«‹β„’π‘,πœ†ξ…ž. If β„±0={βˆ…,Ξ©} also, then 𝐿0βˆžβ«‹β„’π‘,πœ†β«‹β„’π‘,πœ†ξ…žβ«‹πΏ0𝑝.

Proof. (i) By the definition of the sequence (𝑓𝑛)𝑛β‰₯0, we have πΈπ‘›βˆ’1𝑑𝑛𝑓𝐡=π‘ƒπ‘›ξ€Έπœ†ξƒ©π‘ƒξ€·π΅π‘›βˆ’1ξ€Έπ‘ƒξ€·π΅π‘›ξ€ΈπΈπ‘›βˆ’1ξ€Ίπœ’π΅π‘›ξ€»βˆ’πœ’π΅π‘›βˆ’1ξƒͺ=0,(3.13) for every 𝑛β‰₯1. Hence, we obtain that (𝑓𝑛)𝑛β‰₯0 is a martingale.
We next show that the sequence (𝑓𝑛)𝑛β‰₯0 converges in 𝐿𝑝. If limπ‘˜β†’βˆžπ‘ƒ(π΅π‘˜)>0 then the convergence is clear by Remark 3.5. We assume that limπ‘˜β†’βˆžπ‘ƒ(π΅π‘˜)=0. Then, by Lemma 3.3, we can take a sequence of integers 0=π‘˜0<π‘˜1<β‹―<π‘˜π‘—<β‹― that satisfies ξ‚€11+π‘…ξ‚π‘ƒξ‚€π΅π‘˜π‘—ξ‚ξ‚€π΅β‰€π‘ƒπ‘˜π‘—βˆ’1ξ‚ξ‚€π΅β‰€π‘…π‘ƒπ‘˜π‘—ξ‚,(3.14) and π΅π‘˜π‘—βˆ’1=π΅π‘˜ if π‘˜π‘—βˆ’1β‰€π‘˜<π‘˜π‘—. In this case we can write 𝑓𝑛=ξ“π‘˜π‘—β‰€π‘›π‘ƒξ‚€π΅π‘˜π‘—ξ‚πœ†βŽ›βŽœβŽœβŽπ‘ƒξ‚€π΅π‘˜π‘—βˆ’1ξ‚π‘ƒξ‚€π΅π‘˜π‘—ξ‚πœ’π΅π‘˜π‘—βˆ’πœ’π΅π‘˜π‘—βˆ’1⎞⎟⎟⎠.(3.15) Using (3.14) and the assumption πœ†>βˆ’1/𝑝, we have ξ“π‘˜π‘—>π‘›π‘ƒξ‚€π΅π‘˜π‘—ξ‚πœ†β€–β€–β€–β€–β€–π‘ƒξ‚€π΅π‘˜π‘—βˆ’1ξ‚π‘ƒξ‚€π΅π‘˜π‘—ξ‚πœ’π΅π‘˜π‘—βˆ’πœ’π΅π‘˜π‘—βˆ’1β€–β€–β€–β€–β€–πΏπ‘β‰€ξ“π‘˜π‘—>π‘›π‘ƒξ‚€π΅π‘˜π‘—ξ‚πœ†ξ‚΅π‘…β€–β€–πœ’π΅π‘˜π‘—β€–β€–πΏπ‘+β€–β€–πœ’π΅π‘˜π‘—βˆ’1‖‖𝐿𝑝≀2π‘…π‘˜π‘—>π‘›π‘ƒξ‚€π΅π‘˜π‘—ξ‚πœ†+1/𝑝≀2π‘…βˆžξ“π‘—=0ξ‚€11+π‘…ξ‚βˆ’(πœ†+1/𝑝)π‘—π‘ƒξ€·π΅π‘›ξ€Έπœ†+1/π‘ξ€·π΅βˆΌπ‘ƒπ‘›ξ€Έπœ†+1/𝑝.(3.16) Therefore, (𝑓𝑛)𝑛β‰₯0 converges in 𝐿𝑝. We denote by 𝑓 the limit of (𝑓𝑛)𝑛β‰₯0.
We can also deduce from (3.16) that 1π‘ƒξ€·π΅π‘›ξ€Έξ€œπ΅π‘›||π‘“βˆ’πΈπ‘›π‘“||𝑝ξƒͺ𝑑𝑃1/π‘ξ€·π΅β‰²π‘ƒπ‘›ξ€Έπœ†.(3.17) On the other hand, for 𝐡∈𝐴(ℱ𝑛), we have ξ€·π‘“βˆ’πΈπ‘›π‘“ξ€Έπœ’π΅=ξ‚»π‘“βˆ’πΈπ‘›ξ€·π‘“,𝐡=𝐡𝑛,ξ€·0,𝐡≠𝐡𝑛.(3.18)
Combining (3.17) and (3.18), we have β€–π‘“β€–β„’πœ†,𝑝≲1, that is, we get π‘“βˆˆβ„’π‘,πœ†.
(ii) First we show 𝐿0βˆžβ«‹β„’π‘,πœ†β«‹β„’π‘,πœ†β€². By Remark 2.6 we have 𝐿0βˆžβŠ‚β„’π‘,πœ†βŠ‚β„’π‘,πœ†β€². Then we need to show ℒ𝑝,0⧡𝐿0βˆžβ‰ βˆ… and ℒ𝑝,πœ†β€²β§΅β„’π‘,πœ†β‰ βˆ….
We consider 𝑓𝑛 in (3.12) for the sequence π΅π‘˜, π‘˜=0,1,…. Then we can write 𝑓𝑛=ξ“π‘˜π‘—β‰€π‘›βŽ›βŽœβŽœβŽπ‘ƒξ‚€π΅π‘˜π‘—βˆ’1ξ‚π‘ƒξ‚€π΅π‘˜π‘—ξ‚πœ’π΅π‘˜π‘—βˆ’πœ’π΅π‘˜π‘—βˆ’1⎞⎟⎟⎠(3.19) for πœ†=0 and we have 𝑓=(𝑓𝑛)𝑛β‰₯0βˆˆβ„’π‘,0. On the other hand, for a.e. πœ”βˆˆπ΅π‘˜π‘—β§΅π΅π‘˜π‘—+1, 𝑓=π‘“π‘˜π‘—+1=π‘“π‘˜π‘—βˆ’πœ’π΅π‘˜π‘—=𝑗𝑖=0ξƒ©π‘ƒξ€·π΅π‘˜π‘–βˆ’1ξ€Έπ‘ƒξ€·π΅π‘˜π‘–ξ€Έξƒͺ1βˆ’1βˆ’1β‰₯𝑅(𝑗+1)βˆ’1.(3.20)
Then π‘“βˆ‰πΏ0∞ and ℒ𝑝,0⧡𝐿0βˆžβ‰ βˆ….
Next, let 𝑔𝑛=ξ“π‘˜π‘—β‰€π‘›π‘ƒξ‚€π΅π‘˜π‘—ξ‚πœ†ξ…žβŽ›βŽœβŽœβŽπ‘ƒξ‚€π΅π‘˜π‘—βˆ’1ξ‚π‘ƒξ‚€π΅π‘˜π‘—ξ‚πœ’π΅π‘˜π‘—βˆ’πœ’π΅π‘˜π‘—βˆ’1⎞⎟⎟⎠.(3.21) Then 𝑔=(𝑔𝑛)𝑛β‰₯0βˆˆβ„’π‘,πœ†β€². On the other hand, since |||π‘”βˆ’πΈπ‘˜π‘—βˆ’1𝑔|||𝐡=π‘ƒπ‘˜π‘—ξ‚πœ†β€²onπ΅π‘˜π‘—βˆ’1β§΅π΅π‘˜π‘—,(3.22) we have 1π‘ƒξ‚€π΅π‘˜π‘—βˆ’1ξ‚πœ†βŽ›βŽœβŽœβŽ1π‘ƒξ‚€π΅π‘˜π‘—βˆ’1ξ‚ξ€œπ΅π‘˜π‘—βˆ’1|||π‘”βˆ’πΈπ‘˜π‘—βˆ’1𝑔|||π‘βŽžβŽŸβŽŸβŽ π‘‘π‘ƒ1/𝑝β‰₯1π‘ƒξ‚€π΅π‘˜π‘—βˆ’1ξ‚πœ†βŽ›βŽœβŽœβŽ1π‘ƒξ‚€π΅π‘˜π‘—βˆ’1ξ‚ξ€œπ΅π‘˜π‘—βˆ’1β§΅π΅π‘˜π‘—|||π‘”βˆ’πΈπ‘˜π‘—βˆ’1𝑔|||π‘βŽžβŽŸβŽŸβŽ π‘‘π‘ƒ1/𝑝β‰₯π‘ƒξ‚€π΅π‘˜π‘—ξ‚πœ†β€²π‘ƒξ‚€π΅π‘˜π‘—βˆ’1ξ‚πœ†βŽ›βŽœβŽœβŽπ‘ƒξ‚€π΅π‘˜π‘—βˆ’1β§΅π΅π‘˜π‘—ξ‚π‘ƒξ‚€π΅π‘˜π‘—βˆ’1ξ‚βŽžβŽŸβŽŸβŽ 1/𝑝β‰₯π‘…πœ†β€²(1+𝑅)βˆ’πœ†β€²βˆ’1/π‘π‘ƒξ‚€π΅π‘˜π‘—βˆ’1ξ‚πœ†β€²βˆ’πœ†βŸΆβˆžasπ‘—βŸΆβˆž,(3.23) since πœ†ξ…ž<πœ†. This shows ℒ𝑝,πœ†β€²β§΅β„’π‘,πœ†β‰ βˆ….
Finally, if β„±0={βˆ…,Ξ©} also, then by Remark 2.7 and ℒ𝑝,πœ†β€²β§΅β„’π‘,πœ†β‰ βˆ… we have ℒ𝑝,πœ†β«‹πΏ0𝑝. This shows the conclusion.

Remark 3.7. In Proposition 3.6, 𝑓=(𝑓𝑛)𝑛β‰₯0 in (3.12) converges in 𝐿𝑝 as in the above proof. Moreover, the limit belongs to both ℒ𝑝,πœ† and 𝐿𝑝,πœ† when βˆ’1/𝑝<πœ†<0, since we will show that 𝐿𝑝,πœ†=ℒ𝑝,πœ† in the proof of Theorem 3.1. However, it converges in neither ℒ𝑝,πœ† nor 𝐿𝑝,πœ†. Actually, by a similar calculation to 𝑔=(𝑔𝑛)𝑛β‰₯0 in the above proof, we have |||ξ€·π‘“βˆ’π‘“π‘›ξ€Έβˆ’πΈπ‘˜π‘—βˆ’1ξ€·π‘“βˆ’π‘“π‘›ξ€Έ|||=|||π‘“βˆ’πΈπ‘˜π‘—βˆ’1𝑓|||𝐡=π‘ƒπ‘˜π‘—ξ‚πœ†onπ΅π‘˜π‘—βˆ’1β§΅π΅π‘˜π‘—,(3.24) for π‘›β‰€π‘˜π‘—βˆ’1, and then 1π‘ƒξ‚€π΅π‘˜π‘—βˆ’1ξ‚πœ†βŽ›βŽœβŽœβŽ1π‘ƒξ‚€π΅π‘˜π‘—βˆ’1ξ‚ξ€œπ΅π‘˜π‘—βˆ’1|||π‘“βˆ’πΈπ‘˜π‘—βˆ’1𝑓|||π‘βŽžβŽŸβŽŸβŽ π‘‘π‘ƒ1/𝑝β‰₯π‘…πœ†(1+𝑅)βˆ’πœ†βˆ’1/𝑝.(3.25) By Remark 2.5, we have 2β€–β€–π‘“βˆ’π‘“π‘›β€–β€–πΏπ‘,πœ†β‰₯β€–β€–π‘“βˆ’π‘“π‘›β€–β€–β„’π‘,πœ†β‰₯π‘…πœ†(1+𝑅)βˆ’πœ†βˆ’1/𝑝.(3.26) This shows that (𝑓𝑛)𝑛β‰₯0 converges in neither ℒ𝑝,πœ† nor 𝐿𝑝,πœ†.

Proposition 3.8. Let 1≀𝑝<∞.(i)If β„±0={βˆ…,Ξ©} and πœ†β‰€βˆ’1/𝑝, then 𝐿0𝑝=𝐿𝑝,πœ†=𝐿𝑝,πœ†,β„±=ℒ𝑝,πœ†=ℒ𝑝,πœ†,β„± with‖𝑓‖𝐿𝑝,πœ†,β„±=‖𝑓‖𝐿𝑝,πœ†=‖𝑓‖𝐿𝑝≀‖𝑓‖ℒ𝑝,πœ†β‰€β€–π‘“β€–β„’π‘,πœ†,ℱ≀2‖𝑓‖𝐿𝑝,πœ†,β„±.(3.27)(ii)If πœ†β‰₯0, then 𝐿𝑝,πœ†=𝐿𝑝,πœ†,β„±βŠ‚β„’π‘,πœ†=ℒ𝑝,πœ†,β„±βŠ‚πΏ0𝑝 with‖𝑓‖𝐿𝑝≀‖𝑓‖𝐿𝑝,πœ†,β„±=‖𝑓‖ℒ𝑝,πœ†β‰€β€–π‘“β€–πΏπ‘,πœ†,β„±=‖𝑓‖𝐿𝑝,πœ†.(3.28)

Proof. (i) Let πœ†β‰€βˆ’1/𝑝. For any π‘“βˆˆπΏ0𝑝 and any π΅βˆˆβ„±π‘›, 1𝑃(𝐡)πœ†ξ‚΅1ξ€œπ‘ƒ(𝐡)𝐡||𝑓||𝑝𝑑𝑃1/𝑝=𝑃(𝐡)βˆ’πœ†βˆ’1/π‘ξ‚΅ξ€œπ΅||𝑓||𝑝𝑑𝑃1/𝑝≀𝑃(Ξ©)βˆ’πœ†βˆ’1/π‘ξ‚΅ξ€œΞ©||𝑓||𝑝𝑑𝑃1/𝑝=‖𝑓‖𝐿𝑝=𝑃(Ξ©)βˆ’πœ†βˆ’1/π‘ξ‚΅ξ€œΞ©|π‘“βˆ’πΈ0𝑓|𝑝𝑑𝑃1/𝑝.(3.29) Then by the definition of the norms and the assumption β„±0={βˆ…,Ξ©} we see that ‖𝑓‖𝐿𝑝,πœ†,β„±=‖𝑓‖𝐿𝑝,πœ†=‖𝑓‖𝐿𝑝≀‖𝑓‖ℒ𝑝,πœ†β‰€β€–π‘“β€–β„’π‘,πœ†,β„±. By the same observation as Remark 2.5 we have ‖𝑓‖ℒ𝑝,πœ†,ℱ≀2‖𝑓‖𝐿𝑝,πœ†,β„±.
(ii) Let πœ†β‰₯0. By Remark 2.10 we need to show only 𝐿𝑝,πœ†βŠ‚πΏπ‘,πœ†,β„± with ‖𝑓‖𝐿𝑝,πœ†,ℱ≀‖𝑓‖𝐿𝑝,πœ† and ℒ𝑝,πœ†βŠ‚β„’π‘,πœ†,β„±βŠ‚πΏ0𝑝 with ‖𝑓‖𝐿𝑝≀‖𝑓‖ℒ𝑝,πœ†,ℱ≀‖𝑓‖ℒ𝑝,πœ†.
Let π‘“βˆˆπΏπ‘,πœ†. For any π΅βˆˆβ„±π‘›, there exists a sequence of atoms π΅β„“βˆˆπ΄(ℱ𝑛), β„“=1,2,…, such that 𝐡=βˆͺℓ𝐡ℓ and βˆ‘π‘ƒ(𝐡)=ℓ𝑃(𝐡ℓ). Then ξ€œπ΅||𝑓||𝑝𝑑𝑃=β„“ξ€œπ΅β„“||𝑓||π‘ξ“π‘‘π‘ƒβ‰€β„“π‘ƒξ€·π΅β„“ξ€Έπœ†π‘+1‖𝑓‖𝑝𝐿𝑝,πœ†ξ€·π΅β‰€π‘ƒβ„“ξ€Έπœ†π‘+1‖𝑓‖𝑝𝐿𝑝,πœ†,(3.30) since πœ†π‘+1β‰₯1. Therefore π‘“βˆˆπΏπ‘,πœ†,β„± and ‖𝑓‖𝐿𝑝,πœ†,ℱ≀‖𝑓‖𝐿𝑝,πœ†. Similarly, we have ‖𝑓‖ℒ𝑝,πœ†,ℱ≀‖𝑓‖ℒ𝑝,πœ† for π‘“βˆˆβ„’π‘,πœ†. By the definition of ℒ𝑝,πœ†,β„± norm we have ‖𝑓‖𝐿𝑝=β€–π‘“βˆ’πΈ0𝑓‖𝐿𝑝≀𝑃(Ξ©)πœ†+1/𝑝‖𝑓‖ℒ𝑝,πœ†,β„±=‖𝑓‖ℒ𝑝,πœ†,β„± for π‘“βˆˆβ„’π‘,πœ†,β„±.

Proof of Theorem 3.1. (i) We have the conclusion by Proposition 3.8 without the assumption that {ℱ𝑛}𝑛β‰₯0 is regular or that (Ξ©,β„±,𝑃) is nonatomic.
(ii) By Proposition 3.6 we only need to prove 𝐿𝑝,πœ†=ℒ𝑝,πœ† with (1/2)‖𝑓‖ℒ𝑝,πœ†β‰€β€–π‘“β€–πΏπ‘,πœ†β‰€πΆβ€–π‘“β€–β„’π‘,πœ†. The first norm inequality follows from Remark 2.5. We show the second one. Note that we do not need the assumption that β„±0={βˆ…,Ξ©} or that (Ξ©,β„±,𝑃) is nonatomic.
Let π‘“βˆˆβ„’π‘,πœ†. Then, for any 𝐡∈𝐴(ℱ𝑛), ξ‚΅1ξ€œπ‘ƒ(𝐡)𝐡|𝑓|𝑝𝑑𝑃1/𝑝≀1ξ€œπ‘ƒ(𝐡)𝐡|π‘“βˆ’πΈπ‘›π‘“|𝑝𝑑𝑃1/𝑝+||||1ξ€œπ‘ƒ(𝐡)𝐡||||𝑓(πœ”)𝑑𝑃≀𝑃(𝐡)πœ†β€–π‘“β€–β„’π‘,πœ†+||||1π‘ƒξ€œ(𝐡)𝐡||||,𝑓(πœ”)𝑑𝑃(3.31) since 𝐸𝑛1𝑓=ξ€œπ‘ƒ(𝐡)𝐡𝑓(πœ”)𝑑𝑃on𝐡.(3.32) If π΅βˆˆβ„±0, then 𝐸𝑛𝑓=0 on 𝐡. Assume that π΅βˆ‰β„±0. By Lemma 3.3 we can choose π΅π‘˜π‘—βˆˆπ΄(β„±π‘˜π‘—), 0=π‘˜0<π‘˜1<β‹―<π‘˜π‘šβ‰€π‘›, such that π΅π‘˜0βŠƒπ΅π‘˜1βŠƒπ΅π‘˜2βŠƒβ‹―βŠƒπ΅π‘˜π‘š=𝐡 and that (1+1/𝑅)𝑃(π΅π‘˜π‘—)≀𝑃(π΅π‘˜π‘—βˆ’1)≀𝑅𝑃(π΅π‘˜π‘—). Then, since 1π‘ƒξ€·π΅π‘˜0ξ€Έξ€œπ΅π‘˜0𝑓(πœ”)𝑑𝑃=0,(3.33) we have 1ξ€œπ‘ƒ(𝐡)𝐡1𝑓(πœ”)𝑑𝑃=ξ€œπ‘ƒ(𝐡)𝐡1𝑓(πœ”)π‘‘π‘ƒβˆ’π‘ƒξ€·π΅π‘˜0ξ€Έξ€œπ΅π‘˜0=𝑓(πœ”)π‘‘π‘ƒπ‘šξ“π‘—=1βŽ›βŽœβŽœβŽ1π‘ƒξ‚€π΅π‘˜π‘—ξ‚ξ€œπ΅π‘˜π‘—1𝑓(πœ”)π‘‘π‘ƒβˆ’π‘ƒξ‚€π΅π‘˜π‘—βˆ’1ξ‚ξ€œπ΅π‘˜π‘—βˆ’1⎞⎟⎟⎠=𝑓(πœ”)π‘‘π‘ƒπ‘šξ“π‘—=11π‘ƒξ‚€π΅π‘˜π‘—ξ‚ξ€œπ΅π‘˜π‘—ξ‚ƒπ‘“βˆ’πΈπ‘˜π‘—βˆ’1𝑓(πœ”)𝑑𝑃.(3.34) By HΓΆlder’s inequality and the assumption πœ†<0 we have ||||1ξ€œπ‘ƒ(𝐡)𝐡𝑓||||≀(πœ”)π‘‘π‘ƒπ‘šξ“π‘—=1βŽ›βŽœβŽœβŽ1π‘ƒξ‚€π΅π‘˜π‘—ξ‚ξ€œπ΅π‘˜π‘—|||π‘“βˆ’πΈπ‘˜π‘—βˆ’1𝑓|||π‘βŽžβŽŸβŽŸβŽ π‘‘π‘ƒ1/π‘β‰²π‘šξ“π‘—=1βŽ›βŽœβŽœβŽ1π‘ƒξ‚€π΅π‘˜π‘—βˆ’1ξ‚ξ€œπ΅π‘˜π‘—βˆ’1|||π‘“βˆ’πΈπ‘˜π‘—βˆ’1𝑓|||π‘βŽžβŽŸβŽŸβŽ π‘‘π‘ƒ1/π‘β‰€π‘šξ“π‘—=1π‘ƒξ‚€π΅π‘˜π‘—βˆ’1ξ‚πœ†β€–π‘“β€–β„’π‘,πœ†β‰€π‘šξ“π‘—=1ξ‚΅ξ‚€11+Rξ‚π‘šβˆ’π‘—+1π‘ƒξ€·π΅π‘˜π‘šξ€Έξ‚Άπœ†β€–π‘“β€–β„’π‘,πœ†βˆΌπ‘ƒ(𝐡)πœ†β€–π‘“β€–β„’π‘,πœ†.(3.35) Therefore we have ‖𝑓‖𝐿𝑝,πœ†β‰²β€–π‘“β€–β„’π‘,πœ†.
(iii) Let πœ†=0. By Remark 2.6 and Proposition 3.6 we have 𝐿0βˆžβŠ‚πΏπ‘,0 with ‖𝑓‖𝐿𝑝,0β‰€β€–π‘“β€–πΏβˆž and 𝐿0βˆžβ«‹β„’π‘,0.
Next we show 𝐿𝑝,0βŠ‚πΏ0∞ and β€–π‘“β€–πΏβˆžβ‰€β€–π‘“β€–πΏπ‘,0.
Let π‘“βˆˆπΏπ‘,0 and 𝑓≠0 a.e. Take a positive number π‘Ÿ such that 𝑃(|𝑓|>π‘Ÿ)>0. For any πœ–>0, there exists 𝑛 and π΅βˆˆβ„±π‘› such that 𝑃||𝑓||>𝐡∩>π‘Ÿξ€Ύξ€Έ(1βˆ’πœ–)𝑃(𝐡),(3.36) because β„± is generated by βˆͺ𝑛ℱ𝑛. For the above 𝐡, we can take a sequence of atoms π΅β„“βˆˆπ΄(ℱ𝑛), β„“=1,2,…, such that 𝐡=βˆͺℓ𝐡ℓ and βˆ‘π‘ƒ(𝐡)=ℓ𝑃(𝐡ℓ). Hence, by the pigeonhole principle, there exists π΅β€²βˆˆπ΄(ℱ𝑛) such that π‘ƒξ€·π΅ξ…žβˆ©ξ€½||𝑓||>𝐡>π‘Ÿξ€Ύξ€Έ(1βˆ’πœ–)π‘ƒξ…žξ€Έ.(3.37) Therefore, we have ‖𝑓‖𝑝𝐿𝑝,0β‰₯1π‘ƒξ€·π΅ξ…žξ€Έξ€œπ΅β€²||𝑓||𝑝β‰₯1π‘‘π‘ƒπ‘ƒξ€·π΅ξ…žξ€Έξ€œπ΅ξ…žβˆ©{|𝑓|>π‘Ÿ}||𝑓||𝑝β‰₯π‘ƒξ€·π΅π‘‘π‘ƒξ…žβˆ©ξ€½||𝑓||>π‘Ÿξ€Ύξ€Έπ‘ƒξ€·π΅ξ…žξ€Έπ‘Ÿπ‘β‰₯(1βˆ’πœ–)π‘Ÿπ‘.(3.38) This shows that 𝑃(|𝑓|>π‘Ÿ)>0 implies ‖𝑓‖𝐿𝑝,0β‰₯π‘Ÿ. Then we have the conclusion.
By Proposition 3.8 and Remark 2.11 we have that BMO(=β„’1,πœ†)=ℒ𝑝,πœ† with equivalent norms.
(iv) Let πœ†>0. For π‘“βˆˆπΏπ‘,πœ†, we suppose that there exists π‘Ÿ>0 such that 𝑃(|𝑓|>π‘Ÿ)>0. Then, for any πœ–>0, we have the same estimate as (3.37). Moreover, we can decompose 𝐡′ 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 𝐡′ in 𝐴(ℱ𝑛+π‘š) for large enough π‘š such that 𝐡′ satisfies (3.37) and 𝑃(𝐡′)<πœ–. Hence we have ‖𝑓‖𝑝𝐿𝑝,πœ†β‰₯1π‘ƒξ‚€ξ‚‹π΅ξ…žξ‚1+πœ†π‘ξ€œξ‚π΅ξ…ž||𝑓||𝑝𝑑𝑃β‰₯(1βˆ’πœ–)π‘Ÿπ‘π‘ƒξ‚€ξ‚‹π΅ξ…žξ‚πœ†π‘β‰₯(1βˆ’πœ–)π‘Ÿπ‘πœ€πœ†π‘,(3.39) that is, π‘Ÿβ‰€πœ–πœ†/(1βˆ’πœ–)1/𝑝‖𝑓‖𝐿𝑝,πœ†. This contradicts that π‘Ÿ>0. Then 𝑓=0 a.e. and 𝐿𝑝,πœ†={0}. By Proposition 3.6 we have that {0}⫋ℒ𝑝,πœ†.
Finally, by Proposition 3.8 and Remark 2.11 we have that β„’1,πœ†=ℒ𝑝,πœ† with equivalent norms.

Next we prove that 𝐿𝑝,πœ†,ℱ⫋𝐿𝑝,πœ† and ℒ𝑝,πœ†,ℱ⫋ℒ𝑝,πœ† in general by an example.

Proposition 3.9. Let (Ξ©,β„±,𝑃) be as follows: [ξ€·β„±Ξ©=0,1),𝐴𝑛=𝐼𝑛,𝑗=𝑗2βˆ’π‘›,(𝑗+1)2βˆ’π‘›)βˆΆπ‘—=0,1,…,2π‘›ξ€Ύβ„±βˆ’1,(3.40)𝑛𝐴ℱ=πœŽπ‘›ξƒ©ξšξ€Έξ€Έ,β„±=πœŽπ‘›β„±π‘›ξƒͺ,𝑃=theLebesguemeasure.(3.41) If βˆ’1/𝑝<πœ†<0, then 𝐿𝑝,πœ†,ℱ⫋𝐿𝑝,πœ† and ℒ𝑝,πœ†,ℱ⫋ℒ𝑝,πœ†.

Proof. We construct 𝑓 such that π‘“βˆˆπΏπ‘,πœ†β§΅πΏπ‘,πœ†,β„±,π‘“βˆˆβ„’π‘,πœ†β§΅β„’π‘,πœ†,β„±.(3.42)
Step 1. Denote the characteristic function of 𝐼𝑛,𝑗 by πœ’π‘›,𝑗 and let 𝑓𝑛=2π‘›βˆ’1𝑗=0𝑓𝑛+π‘š,2π‘šπ‘—,𝑓𝑛,𝑗𝐼=𝑃𝑛,π‘—ξ€Έπœ†ξ€·πœ’π‘›+1,2π‘—βˆ’πœ’π‘›+1,2𝑗+1ξ€Έ,(3.43) where we choose π‘š such that 𝑃𝐼𝑛+π‘š,0ξ€Έπ‘πœ†+1𝐼≀𝑃𝑛,0ξ€Έ.(3.44)
Note that 𝐼𝑛,𝑗=𝐼𝑛+1,2𝑗βˆͺ𝐼𝑛+1,2𝑗+1 and |𝑓𝑛,𝑗|=𝑃(𝐼𝑛,0)πœ†πœ’π‘›,𝑗.
If π‘˜β‰€π‘›+π‘š, then πΈπ‘˜π‘“π‘›=0, and |π‘“π‘›βˆ’πΈπ‘˜π‘“π‘›|=|𝑓𝑛|=𝑃(𝐼𝑛+π‘š,0)πœ†βˆ‘2π‘›βˆ’1𝑗=1πœ’π‘›+π‘š,2π‘šπ‘—. Hence1π‘ƒξ€·πΌπ‘˜,β„“ξ€Έξ€œπΌπ‘˜,β„“||π‘“π‘›βˆ’πΈπ‘˜π‘“π‘›||𝑝ξƒͺ𝑑𝑃1/𝑝=1𝑃(πΌπ‘˜,β„“)ξ€œπΌπ‘˜,β„“||𝑓𝑛||𝑝ξƒͺ𝑑𝑃1/𝑝𝐼=𝑃𝑛+π‘š,0ξ€Έπœ†ξƒ©βˆ‘π‘—βˆΆπΌπ‘šπ‘—π‘›+π‘š,2βŠ‚πΌπ‘˜,ℓ𝑃𝐼𝑛+π‘š,2π‘šπ‘—ξ€Έπ‘ƒξ€·πΌπ‘˜,β„“ξ€Έξƒͺ1/𝑝.(3.45)
If π‘˜β‰€π‘›, then the number of the elements of {π‘—βˆΆπΌπ‘›+π‘š,2π‘šπ‘—βŠ‚πΌπ‘˜,β„“} is the same as of {π‘—βˆΆπΌπ‘›,π‘—βŠ‚πΌπ‘˜,β„“}. Hence 1𝑃(πΌπ‘˜,β„“)ξ€œπΌπ‘˜,β„“||π‘“π‘›βˆ’πΈπ‘˜π‘“π‘›||𝑝ξƒͺ𝑑𝑃1/𝑝=1π‘ƒξ€·πΌπ‘˜,β„“ξ€Έξ€œπΌπ‘˜,β„“||𝑓𝑛||𝑝ξƒͺ𝑑𝑃1/𝑝𝐼≀𝑃𝑛+π‘š,0ξ€Έπœ†ξƒ©π‘ƒξ€·πΌπ‘›+π‘š,0𝑃𝐼𝑛,0ξ€Έξƒͺ1/𝑝𝐼≀1β‰€π‘ƒπ‘˜,β„“ξ€Έπœ†,(3.46) where we use (3.44) and πœ†<0 for the last two inequalities. If 𝑛<π‘˜β‰€π‘›+π‘š, then the number of the elements of {π‘—βˆΆπΌπ‘›+π‘š,2π‘šπ‘—βŠ‚πΌπ‘˜,β„“} is one at most. Hence 1π‘ƒξ€·πΌπ‘˜,β„“ξ€Έξ€œπΌπ‘˜,β„“||π‘“π‘›βˆ’πΈπ‘˜π‘“π‘›||𝑝ξƒͺ𝑑𝑃1/𝑝=1π‘ƒξ€·πΌπ‘˜,β„“ξ€Έξ€œπΌπ‘˜,β„“||𝑓𝑛||𝑝ξƒͺ𝑑𝑃1/𝑝𝐼≀𝑃𝑛+π‘š,0ξ€Έπœ†ξƒ©π‘ƒξ€·πΌπ‘›+π‘š,0ξ€Έπ‘ƒξ€·πΌπ‘˜,β„“ξ€Έξƒͺ1/π‘ξ€·πΌβ‰€π‘ƒπ‘˜,β„“ξ€Έπœ†,(3.47) where we use βˆ’1/π‘ž<πœ†<0 for the last inequality. In the above, if πΌπ‘˜,β„“=𝐼𝑛+π‘š,0, then the equality holds.
If π‘˜>𝑛+π‘š, then π‘“π‘›βˆ’πΈπ‘˜π‘“π‘›=0 and 1π‘ƒξ€·πΌπ‘˜,β„“ξ€Έξ€œπΌπ‘˜,β„“||𝑓𝑛||𝑝ξƒͺ𝑑𝑃1/𝑝𝐼≀𝑃𝑛+π‘š,0ξ€Έπœ†ξ€·πΌβ‰€π‘ƒπ‘˜,0ξ€Έπœ†.(3.48) Therefore, π‘“π‘›βˆˆπΏπ‘,πœ†βˆ©β„’π‘,πœ†,‖‖𝑓𝑛‖‖𝐿𝑝,πœ†=‖‖𝑓𝑛‖‖ℒ𝑝,πœ†=1.(3.49) On the other hand, for the set ⋃𝐡=2π‘›βˆ’1𝑗=0𝐼𝑛+π‘š,2π‘šπ‘—βˆˆβ„±π‘›+π‘š, ξ‚΅1ξ€œπ‘ƒ(𝐡)𝐡||π‘“π‘›βˆ’πΈπ‘›+π‘šπ‘“π‘›||𝑝𝑑𝑃1/𝑝=ξ‚΅1ξ€œπ‘ƒ(𝐡)𝐡||𝑓𝑛||𝑝𝑑𝑃1/𝑝𝐼=𝑃𝑛+π‘š,0ξ€Έπœ†=(2βˆ’π‘›π‘ƒ(𝐡))πœ†.(3.50)
Therefore, ‖𝑓𝑛‖𝐿𝑝,πœ†,β„±,‖𝑓𝑛‖ℒ𝑝,πœ†,β„±β‰₯2βˆ’π‘›πœ†β†’βˆž as π‘›β†’βˆž.
Step 2. Let 𝑓𝑛 be as in Step 1. If π‘˜<𝑛, then, by the same observation as Step 1 we also have that π‘“π‘›πœ’π‘˜,β„“βˆˆπΏπ‘,πœ†βˆ©β„’π‘,πœ†,β€–β€–π‘“π‘›πœ’π‘˜,ℓ‖‖𝐿𝑝,πœ†=β€–β€–π‘“π‘›πœ’π‘˜,ℓ‖‖ℒ𝑝,πœ†=1.(3.51) Moreover, if π‘˜<𝑛, then, for the set ⋃𝐡=(2π‘›βˆ’1𝑗=0𝐼𝑛+π‘š,2π‘šπ‘—)βˆ©πΌπ‘˜,β„“βˆˆβ„±π‘›+π‘š, ξ‚΅1ξ€œπ‘ƒ(𝐡)𝐡||π‘“π‘›πœ’π‘˜,β„“βˆ’πΈπ‘›+π‘šξ€Ίπ‘“π‘›πœ’π‘˜,β„“ξ€»||𝑝𝑑𝑃1/𝑝=ξ‚΅1ξ€œπ‘ƒ(𝐡)𝐡||π‘“π‘›πœ’π‘˜,β„“||𝑝𝑑𝑃1/𝑝𝐼=𝑃𝑛+π‘š,0ξ€Έπœ†=ξ€·2βˆ’π‘›+