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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.

The purpose of this paper is to introduce Morrey-Campanato spaces of martingales. The Lebesgue space

We consider a probability space

We define Morrey-Campanato spaces as the following: let

We give basic properties of martingale Morrey-Campanato spaces and compare these spaces with martingale Lipschitz spaces introduced by Weisz [

The fractional integrals are very useful tools to analyse function spaces in harmonic analysis. Actually, Hardy and Littlewood [

On the other hand, in martingale theory, Watari [

For a martingale

To prove the boundedness of fractional integrals we use a different method from [

We state notation, definitions, and remarks in the next section and give basic properties of Morrey-Campanato spaces in Section

At the end of this section, we make some conventions. Throughout this paper, we always use

Recall that

For a martingale

It is known that if

Let

In Section

Let

Then functionals

The martingale

Let

Assume that

Assume that

(i) Let

Let

In general, for

If

In general,

By definition, if

By definition and Remark

Let

Our definitions of BMO and

Let

Note that the spaces

By the definitions we have the relations

It is known that, if

We also define weak Morrey spaces.

For

In this section we give basic properties of Morrey and Campanato spaces. The following theorem gives the relation between Morrey and Campanato spaces.

Let

If

If

If

If

We can prove (i) without the assumption that

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

Let

Since

By the lemma we see that there exists

Let

Let

For a sequence

Let

(i) By the definition of the sequence

We next show that the sequence

We can also deduce from (

Combining (

(ii) First we show

We consider

Then

Next, let

Finally, if

In Proposition

Let

If

If

(i) Let

(ii) Let

Let

(i) We have the conclusion by Proposition

(ii) By Proposition

Let

(iii) Let

Next we show

Let

By Proposition

(iv) Let

Finally, by Proposition

Next we prove that

Let

We construct

Denote the characteristic function of

Note that

If

If

If

Therefore,

Let

Let

At the end of this section we prove the relation of

The following is well known for classical Morrey spaces on

If

The first inequality followed from Hölder’s inequality. To show the second inequality, we assume that

It is known as Doob’s inequality that (see for example [

In this section we extend (

Let

Next, take

Then

By (

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

For normed spaces

We state our results in Section

The following is for

Assume that

Let

Let a martingale

If

Assume that

For Morrey norms, one has the following.

Assume that

Note that Theorem

In order to prove (

By the same observation as in Remark

Assume that

For Campanato spaces, one has the following.

Assume that

If

If

Assume that

By Remark

First we show the pointwise estimate

where

Take

where

Hence, we can write

since

Then, for

On the other hand, for

Here, let

If

then, by (

since

choosing

we have by (

If

Therefore, we have (

Next, applying the boundedness of the maximal function (Theorem

for any

Applying the boundedness of the maximal function, we have

Then we have (

Applying the boundedness of the maximal function, we have

Then we have (

Let

Applying the boundedness of the maximal function on

Note that, for

By Hölder’s inequality we have

for all

To prove (

Note that

Since

Therefore, to prove (

Using the inequality

Therefore, we have

We have obtained (

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.

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_{p,λ}spaces

_{ϕ}, the Morrey spaces and the Campanato spaces