#### 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 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 [1β5], 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 such that , where is a nondecreasing sequence of sub--algebras of . Following Weisz [5], we call a stochastic basis. For the sake of simplicity, let . We suppose that every -algebra is generated by countable atoms, where is called an atom (more precisely a -atom), if any with satisfies or . Denote by the set of all atoms in . The expectation operator and the conditional expectation operators relative to are denoted by and , respectively.

We define Morrey-Campanato spaces as the following: let and . For , let and let where is the set of all such that .

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 -algebra 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 - boundedness of the fractional integrals is well known as the Hardy-Littlewood-Sobolev theorem derived from [9β11]. 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 , , and 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 (), 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 relative to , denote its martingale difference by (, with convention ). For , we define the fractional integral of by where is an -measurable function such that Then is a martingale for any martingale , since each is bounded, that is, is a martingale transform introduced by Burkholder [22]. This definition of is an extention of the one in [20, 21] which is for dyadic martingales. We can prove the boundedness of fractional integrals as martingale transforms from to , if and . That is, if a martingale is -bounded, then is -bounded and the following inequality holds: where is a positive constant independent of . Further, we prove the boundedness of fractional integrals 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 -algebra 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, Hedbergβs 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 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 , is dependent on the subscripts. If , we then write or and if , we then write .

#### 2. Notation, Definitions, and Remarks

Recall that is a probability space, and a nondecreasing sequence of sub--algebras of such that . For the sake of simplicity, let . As in Section 1, we always suppose that every -algebra is generated by countable atoms, with denoting by the set of all atoms in . We define the fractional integral as a martingale transform by (1.3).

For a martingale relative to , the maximal function of is defined by

It is known that if , then any -bounded martingale converges in . Moreover, if , , then with is an -bounded martingale and converges to in (see, e.g., [26]). For this reason a function and the corresponding martingale with will be denoted by the same symbol . Note also that . In this case

Let be the set of all martingales such that . For , let be the set of all such that . For any , let . Then is an -bounded martingale in . For this reason we can regard as a subset of .

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

*Definition 2.1. *Let and . For , let
and define

Then functionals and are norms on and , respectively. Note that and are not always trivial set even if and , respectively. This property is different from classical Morrey-Campanato spaces on .

The martingale is said to be -bounded if () and . Similarly, the martingale is said to be -bounded if () and .

Proposition 2.2. *Let . Let and be its corresponding martingale with ().*

(i)*Assume that . If , then is -bounded and
Conversely, if is -bounded, then and
*(ii)*Assume that . If , then is -bounded and
Conversely, if is -bounded, then and
*

*Proof. *(i) Let and . Fix any . If , then
If , then taking such that , we have
Hence
Therefore, and for all . This shows that is a -bounded martingale and
Conversely, from the inequality
it follows that
(ii) Let and . Fix any . If , then
If , then . Hence
Therefore, and for all . This shows that is a -bounded martingale and
Conversely, from the inequality
it follows that

*Remark 2.3. *In general, for (resp., ), does not always converge to in (resp., ). See Remark 3.7.

*Remark 2.4. *If *, *then and for *. *Therefore, if is an -bounded martingale, then it is an -bounded martingale. If *, *from the known result it follows that there exists such that , , and converges to in and a.e. Moreover, we can deduce that , since
The stochastic basis is said to be regular, if there exists a constant such that
holds for all nonnegative martingales .

*Remark 2.5. * In general, with *.* Actually, for any ,
Moreover, if is regular and , then we can prove that with equivalent norms (Theorem 3.1 and Remark 3.2).

*Remark 2.6. * By definition, if , then we have that with and with . If , then with .

*Remark 2.7. *By definition and Remark 2.5, if , then with .

*Definition 2.8. *Let and if .

Our definitions of BMO and 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 and *. *For , let
and define

Note that the spaces and can be defined without the assumption that every -algebra is generated by countable atoms.

*Remark 2.10. *By the definitions we have the relations with and with . If , then we can prove that and with the same norms, respectively (see Proposition 3.8). If , then and in general (see Proposition 3.9).

*Remark 2.11. *It is known that, if is regular and , then with for each (see, e.g., [8]).

We also define weak Morrey spaces.

*Definition 2.12. *For and , let
for measurable functions , and define

#### 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 be regular, and be nonatomic. Let . *(i)*If , then and
*(ii)*If , then and
*(iii)*If , then and
*(iv)*If , then and
*

*Remark 3.2. *We can prove (i) without the assumption that is regular or that is nonatomic. In (ii), we can prove that and without the assumption that or that is nonatomic. To show in (ii) and (iii), we can replace the condition that is nonatomic by a weaker condition as in Proposition 3.6(ii), which follows from the condition that is nonatomic. In (iv), we need the condition that is nonatomic to show .

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

Lemma 3.3. *Let be regular. Then every sequence
**
has the following property: for each ,
**
where is the constant in (2.21).*

*Remark 3.4. *Since is an -atom, we always interpret as the inclusion modulo null sets, that is, means . Therefore, means *. *

*Remark 3.5. *By the lemma we see that there exists such that for all , if and only if *. *

*Proof of Lemma 3.3. *Let . Then . By the regularity we have . This shows . In this case, , since . From the definition of it follows that . Then we have
Next we show or . Suppose that
Then
Therefore,
From the regularity and the inequality above it follows that
This means that .

Proposition 3.6. *Let be regular, and .*

(i)*For a sequence , , let and
Then is a martingale in and in .*(ii)*Let . If there exists a sequence , and , then . If also, then . *

*Proof. *(i) By the definition of the sequence , we have
for every . Hence, we obtain that is a martingale.

We next show that the sequence converges in . If then the convergence is clear by Remark 3.5. We assume that . Then, by Lemma 3.3, we can take a sequence of integers that satisfies
and if . In this case we can write
Using (3.14) and the assumption , we have
Therefore, converges in . We denote by the limit of .

We can also deduce from (3.16) that
On the other hand, for , we have

Combining (3.17) and (3.18), we have , that is, we get .

(ii) First we show . By Remark 2.6 we have . Then we need to show and .

We consider in (3.12) for the sequence , . Then we can write
for and we have . On the other hand, for a.e. ,

Then and .

Next, let
Then . On the other hand, since
we have
since . This shows .

Finally, if also, then by Remark 2.7 and we have . This shows the conclusion.

*Remark 3.7. *In Proposition 3.6, in (3.12) converges in as in the above proof. Moreover, the limit belongs to both and when , since we will show that in the proof of Theorem 3.1. However, it converges in neither nor . Actually, by a similar calculation to in the above proof, we have
for , and then
By Remark 2.5, we have
This shows that converges in neither nor .

Proposition 3.8. *Let .*(i)*If and , then with*(ii)*If , then with*

*Proof. *(i) Let . For any and any ,
Then by the definition of the norms and the assumption we see that . By the same observation as Remark 2.5 we have .

(ii) Let . By Remark 2.10 we need to show only with and with .

Let . For any , there exists a sequence of atoms , , such that and . Then
since . Therefore and . Similarly, we have for . By the definition of norm we have for .

*Proof of Theorem 3.1. *(i) We have the conclusion by Proposition 3.8 without the assumption that is regular or that is nonatomic.

(ii) By Proposition 3.6 we only need to prove with . The first norm inequality follows from Remark 2.5. We show the second one. Note that we do not need the assumption that or that is nonatomic.

Let . Then, for any ,
since
If , then on . Assume that . By Lemma 3.3 we can choose , , such that and that . Then, since
we have
By HΓΆlderβs inequality and the assumption we have
Therefore we have .

(iii) Let . By Remark 2.6 and Proposition 3.6 we have with and .

Next we show and .

Let and a.e. Take a positive number such that . For any , there exists and such that
because is generated by . For the above , we can take a sequence of atoms , , such that and . Hence, by the pigeonhole principle, there exists such that
Therefore, we have
This shows that implies . Then we have the conclusion.

By Proposition 3.8 and Remark 2.11 we have that with equivalent norms.

(iv) Let . For , we suppose that there exists such that . Then, for any , we have the same estimate as (3.37). Moreover, we can decompose in (3.37) to atoms in and we have the same estimate as (3.37) for some atom in . Therefore, we can take an atom in for large enough such that satisfies (3.37) and . Hence we have
that is, . This contradicts that . Then a.e. and . By Proposition 3.6 we have that .

Finally, by Proposition 3.8 and Remark 2.11 we have that with equivalent norms.

Next we prove that and in general by an example.

Proposition 3.9. *Let be as follows:
**
If , then and . *

*Proof. *We construct such that
*Step 1. *Denote the characteristic function of by and let
where we choose such that

Note that and .

If , then , and . Hence

If , then the number of the elements of is the same as of . Hence
where we use (3.44) and for the last two inequalities. If , then the number of the elements of is one at most. Hence
where we use for the last inequality. In the above, if , then the equality holds.

If , then and
Therefore,
On the other hand, for the set ,

Therefore, as . *Step 2. *Let be as in Step 1. If , then, by the same observation as Step 1 we also have that
Moreover, if , then, for the set ,
Therefore, .*Step 3. *Let be as in Step 1 and let
Note that . Then
On the other hand, by the same observation as Step 2, we have that
for all . This shows (3.42).

At the end of this section we prove the relation of and . Recall that

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

Proposition 3.10. *If and , then
*

*Proof. *The first inequality followed from HΓΆlderβs inequality. To show the second inequality, we assume that and prove that . From , we have . For any atom , let and
Then
Since we have
We get the conclusion.

#### 4. Maximal Function

It is known as Doobβs inequality that (see for example [5, Pages 20-21])

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 .

Theorem 4.1. *Let and . Then, for ,
*

*Proof. **Case 1* (). For any and , let and . Then, using (4.1), we have