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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.

Lebesgue spaces

In martingale theory, Weisz [

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 [

On these martingale spaces, we introduce generalized fractional integrals as martingale transforms and prove their boundedness. Our result extends several results in [

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

Let

The expectation operator and the conditional expectation operators relative to

Let

Let

In this paper, we always postulate the following condition on

Let

By the condition (

Let

In general,

For

If

A function

For the case

Recall that

In this paper, we do not always assume that each sub-

Next we define Morrey-Hardy and Campanato-Hardy spaces, based on square functions, with respect to

Let

By (

If we take

In the end of this section, we present the definition of regularity on

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 Section

First we state basic properties of the norms.

Let

Let

For

The following is well known as Burkholder’s inequality.

If

For expressions of the constants

Our first result is the following, which is an extension of Burkholder’s inequality to martingale Campanato spaces.

Let

Next we give the relations between

Let

We give a relation between martingale Morrey spaces and martingale Campanato spaces in the following form.

Suppose that every

Using Theorems

Suppose that every

For the martingale

Let

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 Section

Let

We now define a generalized fractional integral

Suppose that every

For quasinormed spaces

We first study the boundedness on the spaces

In Theorem

Let

If every

Assume that every

We next study the boundedness on martingale Morrey-Hardy spaces

Recall that

Let

According to Proposition

In this case, if

Suppose that every

As a consequence of Theorem

Suppose that every

The following extends the results for dyadic martingales in [

Suppose that every

We prepare some lemmas to prove the results in Sections

Let

Let

Suppose that every

Suppose that every

Let

Let

In Theorem

Let

Note that this lemma is the counterpart to the technique in [

By the definition of

Using the doubling condition on

In the proof of Theorem

Suppose that every

Let

In the course of the proof, the embedding

In this section, we prove the results in Section

Proposition

Recall that

We first show Theorem

Let

We next show (

We next show Theorem

Inequality (

We now show (

Suppose that

From (

By the same way as above, we have (

We now prove Theorem

The part

Let

Let

If

We will now prove Theorem

We may assume that

By Hölder’s inequality and Theorem

Recall the notation

In this section, we prove the results in Section

Recall that

Using the assumption that

Let

Assume that (

We first show the part

On

We now show (

We now show the part

Let

If

In the light of Remark

In words of harmonic analysis, this corresponds to the embedding

In this subsection, we prove Proposition

To prove

To prove (

Suppose that we can choose

For the set

Since

Therefore, we have

For

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are thankful to Professor Masaaki Fukasawa in Osaka University for his kind hint about Remarks