Fractional integrals on martingale spaces (The structure of function spaces and its environment)

$\Gamma$

ractional

_integrals

on

martingale

_spaces

茨城大学

理学部

中井英一

_(Eiichi

_Nakai)

Department

of

_Mathematics)

Ibaraki

_University

大阪教育大学

教育学部 貞末 岳

_{(Gaku Sadasue)}

Department

of

_Mathematics)

Osaka

_Kyoiku

_University

1

Introduction

In this paper, we review known results on fractional

integrals

of

martingales

and

state some newresults. We introduce commutator of fractional

integral

ofmartin‐

gales,

andstateacharacterizationof

Lipschitz martingales by

boundedness of these

commutatorson

martingale Morrey

_spaces. We alsostatea

_property

of

sharp

func‐ tions on

martingale Morrey

_spaces. This_paper isan announcement of the authors’

recent results

_[11].

Let

_{( $\Omega$, \mathcal{F}, P)}

be a

probability

_space and let

\{\mathcal{F}_{n}\}_{n\geq 0}

be a

nondecreasing

se‐

quence of

sub‐a‐algebras

of \mathcal{F} such that \mathcal{F}=

$\sigma$(\displaystyle \bigcup_{n}\mathcal{F}_{n})

. We suppose that every

$\sigma$

‐algebra

\mathcal{F}_{n}

is

generated by

countable atoms, where B \in

_{\mathcal{F}_{n}}

is called an atom

(more

precisely

\mathrm{a}(\mathcal{F}_{n}, P)‐atom),

if_any A\subset B with

_{A\in \mathcal{F}_{n}}

satisfies

_P(A)=P(B)

or

P(A)

=0. Denote

by

A(\mathcal{F}_{n})

the set of all atoms in

\mathcal{F}_{n}

. We also suppose that

( $\Omega$, \mathcal{F}, P)

is non‐atomic.

The

_{expectation operator}

is denoted

_by

E. Let

L_{p,1\mathrm{o}\mathrm{c}}

be the set of all measur‐

able functions such that

_{|f|^{p}$\chi$_{B}}

is

_integrable

for all B \in

A(\mathcal{F}_{0})

. If

\mathcal{F}_{0}

=

\{ $\Omega$, \emptyset\},

2000 Mathematics _{Subject Classification. Primary 46\mathrm{E}30,}60\mathrm{G}46; Secondary42\mathrm{B}35, 26\mathrm{A}33.

Keywords andphrases. martingale,fractionalintegral, commutator,Morrey‐Campanatospace.

The first author was _{supported by} Grant‐in‐Aid for Scientific Research

(B),

No. _{15\mathrm{H}03621,}

Japan Societyfor the Promotion of Science. The second author wassupported by Grant‐in‐Aid

then

_{L_{p,1\mathrm{o}\mathrm{c}}}

=

L_{p}

. An

\mathcal{F}_{n}

‐measurable function 9 \in

L_{1,1\mathrm{o}\mathrm{c}}

is called the conditional

expectation

of

_{f\in L_{1,1\mathrm{o}\mathrm{c}}}

relative to

_{\mathcal{F}_{n}}

if

E[g$\chi$_{B}$\chi$_{G}]=E[f$\chi$_{B}$\chi$_{G}]

for all

_{B\in A(\mathcal{F}_{0})}

and

G\in \mathcal{F}_{n}.

We denote

_by

_{E_{n}f}

the conditional

expectation

of

_f

relative to

_{\mathcal{F}_{n}}

. We say a

sequence

(f_{n})_{n\geq 0}

in

L_{1,1\mathrm{o}\mathrm{c}}

is a

martingale

relative to

\{\mathcal{F}_{n}\}_{n\geq 0}

if it is

adapted

to

\{\mathcal{F}_{n}\}_{n\geq 0}

and satisfies

_{E_{n}[f_{m}]=f_{n}}

for every n\leq m.

2

Definitions

ラ

notation and

known results

In this

_section,

we

give

definitions and recall known results.

We first recall the definition of

_martingale

_Morrey

spaces

L_{\mathrm{p}, $\lambda$}

and

martingale

Campanato

spaces

\mathcal{L}_{p, $\lambda$}.

Definition 2.1. Let p\in

_{[1, \infty)}

and

_{$\lambda$\in(-\infty, \infty)}

. For

f\in L_{1,1\mathrm{o}\mathrm{c}}

, let

\displaystyle \Vert f\Vert_{L_{p, $\lambda$}}=\sup_{n\geq 0}\sup_{B\in A(\mathcal{F}_{n})}\frac{1}{P(B)^{ $\lambda$}} (\frac{1}{P(B)}\int_{B}|f|^{p}dP)^{1/p}

\displaystyle \Vert f\Vert_{\mathcal{L}_{p, $\lambda$}}=\sup_{n\geq 0}\sup_{B\in A(\mathcal{F}_{n})}\frac{1}{P(B)^{ $\lambda$}} (\frac{1}{P(B)}\int_{B}|f-E_{n}f|^{p}dP)^{1/p}

and define

L_{p, $\lambda$}=\{f\in L_{p,1\mathrm{o}\mathrm{c}}: \Vert f\Vert_{L_{p, $\lambda$}} <\infty\}, \mathcal{L}_{p, $\lambda$}=\{f\in L_{p,1\mathrm{o}\mathrm{c}}: \Vert f\Vert_{\mathcal{L}_{p, $\lambda$}} <\infty\}.

Then functionals

_{\Vert f\Vert_{L_{p, $\lambda$}}}

and

_{\Vert f\Vert_{\mathcal{L}_{p, $\lambda$}}}

are norms.

We

_{regard martingale}

BMO spaces and

martingale Lipschitz

spaces as

special

classes of

_{martingale Campanato}

spaces.

Definition 2.2. Let BMO

_{=\mathcal{L}_{1,0}}

and

_Lip

_{( $\delta$)=\mathcal{L}_{1, $\delta$}}

for $\delta$>0.

The filtration

_{\{\mathcal{F}_{n}\}_{n\geq 0}}

is said to be

_regular,

if there exists a constant R _\geq 2

suchthat

(2.1)

f_{n}\leq Rf_{n-1}

holds for all

_nonnegative

_martingales

_{(f_{n})_{n\geq 0}.}

Theorem 2.1. Assume that

_{\{\mathcal{F}_{n}\}_{n\geq 0}}

is

_regular.

Let

_1<p<\infty

.

Then,

\Vert f\Vert_{\mathrm{B}\mathrm{M}\mathrm{O}}\sim\Vert f\Vert_{\mathcal{L}_{p,0}}

and

_{\Vert f\Vert_{\mathrm{L}\mathrm{i}\mathrm{p}( $\delta$)}\sim\Vert f\Vert_{\mathcal{L}_{p, $\delta$}}.}

Fractional

_integrals

for

_martingales

was first introduced

by

Chao and Ombe

[3]

as follows.

Definition2.3

_([3]).

Let $\alpha$>0. Fora

dyadic martingale

f=(f_{n})_{n\geq 0}

, itsfractional

integral

_{I_{ $\alpha$}f=((I_{ $\alpha$}f)_{n})_{n\geq 0}}

is defined

_by

(I_{ $\alpha$}f)_{n}=\displaystyle \sum_{k=0}^{n}2^{-k $\alpha$}(f_{k}-f_{k-1})

.

Later,

I_{ $\alpha$}

is defined for more

general martingales.

Recall our

assumption

that

every $\sigma$

‐algebra

\mathcal{F}_{n}

is

generated by

countable atoms. In

_[9],

_{I_{ $\alpha$}}

is defined for this case.

Let

(2.2)

$\beta$_{n}=\displaystyle \sum_{B\in A(\mathcal{F}_{n})}P(B)$\chi$_{B}

) n=0,

1, 2,

Definition 2.4

_([9]).

Let $\alpha$ > 0. For a

martingale f

=

(f_{n})_{n\geq 0}

, its fractional

integral

I_{ $\alpha$}f=((I_{ $\alpha$}f)_{n})_{n\geq 0}

is defined

_by

(I_{ $\alpha$}f)_{n}=\displaystyle \sum_{k=0}^{n}$\beta$_{k-1}^{ $\alpha$}(f_{k}-f_{k-1})

.

with convention

_{$\beta$_{-1}=$\beta$_{0}}

and

_{f_{-1}=0.}

In above two

_definitions,

_{I_{ $\alpha$}}

is defined on

martingale

_spaces. In this _paper, we define

I_{ $\alpha$}

on function_spaces.

Definition 2.5. Let $\alpha$>0. For

f\in

L_{1,1\mathrm{o}\mathrm{c}}

, its fractional

integral I_{ $\alpha$}f

with respect

to

_{\{\mathcal{F}_{n}\}_{n\geq 0}}

is defined

_by

(2.3)

I_{ $\alpha$}f=\displaystyle \sum_{k=0}^{\infty}$\beta$_{k-1}^{ $\alpha$}

(Ekf—

E_{k-1}f

)

Remark2.1. Asisshown in

_[9],

theseries

_{$\chi$_{B}\displaystyle \sum_{k=0}^{\infty}($\beta$_{k-1})^{ $\alpha$}(E_{k}f-E_{k-1}f)}

converges in

_{L_{1}}

for every

B\in A(\mathcal{F}_{0})

and

f\in L_{1,1\mathrm{o}\mathrm{c}}

.

Moreover,

E_{n}[I_{ $\alpha$}f]=\displaystyle \sum_{k=0}^{n}$\beta$_{k-1}^{ $\alpha$}(E_{k}f-E_{k-1}f)

.

We recall the

_following

result on the boundedness of

I_{ $\alpha$}.

Theorem 2.2. Assume that

_{\{\mathcal{F}_{n}\}_{n\geq 0}}

is

_regular.

Let 1 _<p< q < \infty, $\alpha$ > 0 and

-1/p\leq $\lambda$<0

.

If

$\alpha$+ $\lambda$<0 and

$\alpha$=1/p-1/q

, then there exists a

positive

constant

C

_{depending only}

onR and $\alpha$ such that

\Vert I_{ $\alpha$}f\Vert_{L_{q, $\alpha$+ $\lambda$}} \leq C\Vert f\Vert_{L_{p, $\lambda$}}.

Remark 2.2. Theorem 2.2 extends

_[3,

Theorem

_1]

in several _ways: from

_dyadic

martingales

to more

general martingales,

from

L_{p}

_{spaces to}

Morrey

_spaces.

Further,

werecall the definition of

generalized

fractional

integrals

of

martingales,

and the definition of

_{generalized Morrey}

spaces.

Definition 2.6. Let

_{($\gamma$_{n})_{n\geq 0}}

bea

_{non‐increasing}

_sequenceof

non‐negative

bounded functions

_adapted

to

_{\{\mathcal{F}_{n}\}_{n\geq 0}}

. For a

martingale

(f_{n})_{n\geq 0}

, its

generalized

fractional

integral

_{I_{ $\gamma$}f=((I_{ $\gamma$}f)_{n})_{n\geq 0}}

is defined as a

martingale by

(I_{ $\gamma$}f)_{n}=\displaystyle \sum_{k=0}^{n}$\gamma$_{k-1}(f_{k}-f_{k-1})

withconvention $\gamma$_{-1}=$\gamma$_{0} and

_{f_{-1}=0.}

Definition 2.7. For p\in

[1, \infty)

and

_$\phi$

:

(0

)1

] \rightarrow(0, \infty)

, let

L_{\mathrm{p}, $\phi$}=\{f\in L_{p,1\mathrm{o}\mathrm{c}}: \Vert f\Vert_{L_{p, $\phi$}}<\infty\},

where

\displaystyle \Vert f\Vert_{L_{p, $\phi$}}=\sup_{n\geq 0}\sup_{B\in A(\mathcal{F}_{n})}\frac{1}{ $\phi$(P(B))} (\frac{1}{P(B)}\int_{B}|f|^{p}dP)^{1/p}

Boundedness of

_generalized

fractional

_integrals

is studied

_extensively

in

_[10].

Theorem 2.3

_([10]).

Let 1 _{<p<q<\infty and}

_$\phi$

:

(0,1] \rightarrow (0, \infty)

. Assume that

$\phi$

is almost

_decreasin9.

_If

there exists a

positive

constant C such that

(2.4)

\displaystyle \sum_{k=0}^{n}($\gamma$_{k-1}-$\gamma$_{k}) $\phi$(b_{k})+$\gamma$_{n} $\phi$(b_{n})\leq C $\phi$(b_{n})^{p/q}

for

all

_{n\geq 0}

with _{convention $\gamma$_{-1}=$\gamma$_{0}}, then

I_{ $\gamma$}

is bounded

from

L_{\mathrm{p}, $\phi$}

to

L_{q,$\phi$^{p/q}}.

3

Some

new

properties:

fractional maximal func‐

tions, sharp

functions and

commutators.

In this

_section,

we state new

_properties

of fractional maximal

functions, sharp

functions and commutators. The

_proofs

of these

_properties

will be

_given

in

_[11].

For

_{f\in L_{1}}

its fractional maximalfunction

_{M_{ $\alpha$}f}

is defined

_by

(3.1)

_{M_{ $\alpha$}f=\displaystyle \sup_{n\geq 0}($\beta$_{n})^{ $\alpha$}|E_{n}f|.}

Using

Theorem 2.2 and the

_positivity

of

I_{ $\alpha$}

, wehave the

following

theorem.

Theorem3.1. Assume that

_{\{\mathcal{F}_{n}\}_{n\geq 0}}

is

_regular.

Let

_{1<p<q<\infty and-1/p+ $\alpha$=}

-1/q

. Then

M_{ $\alpha$}

is bounded

from

L_{p, $\lambda$}

to

L_{q, $\alpha$+ $\lambda$}.

Wenext recall the definition of

_sharp

functions.

(3.2)

_{M^{\#}f=\displaystyle \sup_{n\geq 0}E_{n}[|f-E_{n-1}f|].}

The

_following

theorem is well‐known. See

_[16]

and

_[6].

Theorem 3.2. Let

1<p<\infty

.

Then,

there exists a

positive

constant C

depending

only

on_p such that

\Vert f\Vert_{L_{p}}\leq C\Vert M^{\#}f\Vert_{L_{p}}.

Our resulton

sharp

functions is to

give

anextension of Theorem 3.2 to martin‐

gale Morrey

spaces.

Theorem 3.3. Assume that

_{\{\mathcal{F}_{n}\}_{n\geq 0}}

is

_regular.

Let

_{1<p_{0}<p<\infty}

_and-1/p\leq

$\lambda$<0.

If

Mf\in L_{p_{0}, $\lambda$}

, then

(3.3)

_{\Vert f\Vert_{L_{p, $\lambda$}} \leq C_{p, $\lambda$,R}\Vert M^{\#}f\Vert_{L_{\mathrm{p}, $\lambda$}},}

where

_{C_{p, $\lambda$,R}}

is a

positive

constant

depending only

on_p, $\lambda$ and R in

(2.1).

Toshow Theorem

_3.3,

we use a

good

$\lambda$

‐inequality

which is a

martingale

version

of

_{Komori‐Furuya’s}

result in

_[5].

Wenowintroducecommutators. Let

_{p\geq 1}

and let

_p'

be the

_conjugate

exponent

ofp. If

f\in L_{p,1\mathrm{o}\mathrm{c}}

and

b\in L_{p',1\mathrm{o}\mathrm{c}}

, then the commutator

iswell‐defined.

In

_[3],

Chao and Ombe showed the

_following

characterization theorem for

_dyadic

BMO‐martingales.

Theorem3.4

_([3]).

Let

_1<p<q<\infty

and

_{$\alpha$=1/p-1/q}

. Let

I_{ $\alpha$}

be the

fractional

integral defined

in

_Defintition

2.3.

_Then,

b

_belongs

to

_dyadic

BMO space

if

and

only

if

the commutator

_{[b, I_{ $\alpha$}]}

is bounded

_from

_{L_{p}}

to

_{L_{q}.}

We extend Theorem 3.4 to the

_following

theorem.

Theorem 3.5. Assume that

_{\{\mathcal{F}_{n}\}_{n\geq 0}}

is

_regular.

Let

_{1<p<q<\infty,}

_{$\alpha$=1/p-1/q,}

$\delta$\geq 0

and $\lambda$ <0.

Suppose

that $\delta$+ $\alpha$+ $\lambda$<0.

Then,

b\in \mathrm{L}\mathrm{i}\mathrm{p}( $\delta$)

, b\in BMO when

$\delta$=0,

if

and

only if

the commutator

[b, I_{ $\alpha$}]

is bounded

from

L_{p, $\lambda$}

to

L_{q, $\delta$+ $\alpha$+ $\lambda$}.

Remark 3.1. In Theorem

_3.5,

we extend Theorem 3.4 inseveral _ways. We extend Theorem 3.4 from

_{dyadic martingales}

to more

general martingales,

from BMO‐

martingales

to

_{Lipschitz martingales}

and from

_{L_{p}}

_{spaces to}

_Morrey

spaces.

The

_proof

of Theorem 3.5 consists of theuse of Theorem 3.3 and some _compu‐

tations. The detailed

_proof

will be

_given

in

_[11].

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