Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2012, Article ID 673929,29 pages doi:10.1155/2012/673929

Research Article

Martingale Morrey-Campanato Spaces and Fractional Integrals

Eiichi Nakai1 and Gaku Sadasue2

1 Department of Mathematics, Ibaraki University, Mito, Ibaraki 310-8512, Japan

2 Department of Mathematics, Osaka Kyoiku University, Kashiwara, Osaka 582-8582, Japan

Correspondence should be addressed to Eiichi Nakai, enakai@mx.ibaraki.ac.jp Received 3 March 2012; Revised 23 April 2012; Accepted 23 April 2012 Academic Editor: Dachun Yang

Copyright © 2012 E. Nakai and G. Sadasue. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 (Q, F,P) such that F = o(Un Fn), where {Fn}n>0 is a nondecreasing sequence of sub-o-algebras of F. Following Weisz [5], we call {Fn}n>0 a stochastic basis. For the sake of simplicity, let F-i = F0. We suppose that every o-algebra Fn is generated by countable atoms, where B e Fn is called an atom (more precisely a (Fn,P)-atom), if any A c B with A e Fn satisfies P(A) = P(B) 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 p e [1, to) and X e (-to, to). For f e Li, let

\\f\\L = sup sup |f\pdp)1/P

117 "LP'X n>0 BeA(Fn) P(B)X\P(B) JbU ' /

1/ (1-1) 1 / 1 f \1/P

L- = sis Besu(Fn)W^J B lf - Enf ^^

and let

LpX = {f e LP : \\f \\ v < TO' LP,x = {f e LOp : \\f U^ < to}, (1-2)

where Lp is the set of all f e Lp 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 aalgebra 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-Lq 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 Lipa 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)n>0 relative to {Fn}n>0, denote its martingale difference by dnf = fn — fn- 1 (n > 0, with convention dof = 0). For a > 0, we define the fractional integral Iaf =((Iaf )n)n>0 of f by

(fn = X bh df> (1.3)

where bk is an Fk-measurable function such that

bk(v) = P(B) for a.e. w e B with B e A(Fk)•

Then ((Iaf)n)n>0 is a martingale for any martingale f = (fn)n>0, since each bk is bounded, that is, Ia is a martingale transform introduced by Burkholder [22]. This definition of Ia is an extention of the one in [20, 21] which is for dyadic martingales. We can prove the boundedness of fractional integrals Ia as martingale transforms from Lp to Lq, if 1 <p < q < to and -1/p + a = -1/q. That is, if a martingale f = (fn)n>0 is Lp-bounded, then ((Iaf)n)n>0 is Lq-bounded and the following inequality holds:

supUCfaf)n|L < C sup||fn||Lp, (1.5)

n>0 q n>0 p

where C is a positive constant independent of f. Further, we prove the boundedness of fractional integrals Ia 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 c-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 C 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 Cp, is dependent on the subscripts. If f < Cg, we then write f < g or g > f and if f < g < f, we then write f ~ g.

2. Notation, Definitions, and Remarks

Recall that (Q, F,P) is a probability space, and {Fn}n>0 a nondecreasing sequence of sub-c-algebras of F such that F = c(Un Fn). For the sake of simplicity, let F-i = F0. As in Section 1, we always suppose that every c-algebra Fn is generated by countable atoms, with denoting by A(Fn) the set of all atoms in Fn. We define the fractional integral Ia as a martingale transform by (1.3).

For a martingale f = (fn)n>0 relative to {Fn}n>0, the maximal function f* of f is defined

fn = sup \fm\, f* = sup|fn|- (2.1)

0<m<n n>0

It is known that if p e (1, to), then any Lp-bounded martingale converges in Lp. Moreover, if f e Lp, p e [1, to), then (fn)n>0 with fn = Enf is an Lp-bounded martingale and converges to f in Lp (see, e.g., [26]). For this reason a function f e L1 and the corresponding martingale (fn)n>0 with fn = Enf will be denoted by the same symbol f. Note also that llf 11 Lp = supn>0 II Enf 11 Lp. In this case

fn = sup \Emf\, f* = sup\Enf \ for f e L1. (2.2)

0<m<n n>0

Let M be the set of all martingales such that f0 = 0. For p e [1, to], let Lp be the set of all f e Lp such that Eof = 0. For any f e L0p, let fn = Enf. Then (fn) n>0 is an Lp-bounded martingale in M. For this reason we can regard Lp as a subset of M.

In Section 1 we have introduced Morrey spaces Lp,\ and Campanato spaces Lpr1 as the following.

Definition 2.1. Let p e [1, to) and 1 e (-to, to). For f e L1, let

=sup .ex P5?(m llf i'1*)

1 / 1 c \ 1/p

U 1 = sup sup -—Apif - Enf \pdP) ,

p,i n>0 BeA(Fn) P(B)1 V1 (B) J B /

and define

V = f e L°p : llf ILp, < to} , £pi = f e lp : ||f H^pi < to}• (2.4)

Then functionals ||f \\Lp1 and ||f ||lp1 are norms on Lpr1 and Lp/1, respectively. Note that Lpi and Lpi are not always trivial set {0} even if 1> 0 and 1> 1, respectively. This property is different from classical Morrey-Campanato spaces on Rn.

The martingale f = (fn)n>0 is said to be Lp/1-bounded if fn e Lpr1 (n > 0) and supn>0||fn|lv < to. Similarly, the martingale f = (fn)n>0 is said to be Lp ^-bounded if fn e Lp x (n > 0) and supn>0WfnWLpx < to.

Proposition 2.2. Let 1 < p < to. Let f e L1 and (fn)n>0 be its corresponding martingale with fn = Enf (n > 0).

(i) Assume that 1 e (-to, 0]. If f e Lpr1, then (fn)n>0 is Lpr1-bounded and

llf 11 Lp ,1 > sup 11 fn 11 Lpx • (2.5)

Conversely, if (fn)n>0 is Lp ¿-bounded, then f e Lp,x and

Hf HLp < sup 11 fn 11 Lpx • (2.6)

(ii) Assume that 1 e (-to, to). If f e Lpr1, then (fn)n>0 is Lp ¿-bounded and

HfHLpx > sup 11 fn 11 Lpr1 • (2.7)

Conversely, if (fn)n>0 is Lp^-bounded, then f e Lp 1 and

,p,< ^PWfnWjpy (2.8)

Proof. (i) Let f e LP/x and n > 0. Fix any B e A(Fk)• If k < n, then

(fB |Enf |'dP)< (|B£n[|f|p - ( jB\f \pdp)1/P < P (B)A+1/p||f \\lv y (2.9)

If n < k, then taking Bn e A(Fn) such that B c Bn, we have

P (Bn)

enf - pB)ijdp on ^ (Z10)

p^Z B ^\'dP)"' -( P(Bn> i, iE»f I"dp)

Enil f IHdP

f En [| f |']

P(Bn) Jb„ / (2.11)

w x v'

P(B) iBnlf ^ < P(Bn)A||f ||V < P(B)ALlIL',

Therefore, fn = Enf e Lp/X and Ifnlip^ < If IIl^ for all n > 0. This shows that (fn)n>0 is a Lp ^-bounded martingale and

^M^p, < ||f 11Lp . (2.12)

Conversely, from the inequality

f \f\pdP < liming \Enf \pdP VB e MA(Fm), (2.13)

Jb n^to Jb m

it follows that

U ЦLp x < sup|Enf ЦLp x. (2.14)

(ii) Let f e Lpx and n > 0. Fix any B e A(Fk). If k < n, then

Q |Enf - Ek [Enf]\pd^j1/P = Q | Enf - En[Ekf]\pd^j

<(IßEn^f -Ekf ^]dP)

B (2.15)

kf| dP

(Lf -Ef i

< p (s)a+"'^ i il,..,

If n < k, then Ek [Enf ] = Enf. Hence

(■miBlEnf - Ek Enf ]\VdP) yv = 0. (Z16)

Therefore, fn = Enf e Lp x and Hfn\Up,x < llf \UM for all n > 0. This shows that (fn)n>0 is a x-bounded martingale and

supllfnLp, < llf IIlx (2.17)

Conversely, from the inequality

\ \f - fk\pdP < liminff \ En f - fk]\pdP VB e A(Fk) , k = 1,2..........(2.18)

jb nJb

it follows that

< sup|Enf (2.19)

Remark 2.3. In general, for f e Lp,x (resp., Lp,x), Enf does not always converge to f in Lpx (resp., LpX). See Remark 3.7.

Remark 2.4. If Fo = (0, A), then Lprx c Lp and ||f \\lp < \\f \\lm for f e Lp,x. Therefore, if (fn)n>0 is an Lp,x-bounded martingale, then it is an Lp-bounded martingale. If 1 < p < to, from the known result it follows that there exists f e Lp such that Enf = fn, n > 0, and (fn)n>0 converges to f in Lp and a.e. Moreover, we can deduce that f e Lp,x, since

( f \fn(w)\pdp) 1/P < P (B)X+1/pllfnllL x < P (B)X+1/psupllfnllLp ^ B e (J A(Fm)-

\JB / n>0 m

(2.20)

The stochastic basis (Fn)n>0 is said to be regular, if there exists a constant R > 2 such that

fn < Rfn-1 (2.21)

holds for all nonnegative martingales (fn)n>0.

Remark 2.5. In general, Lprx c Lprx with ||f \\lpx < 2\\f HiM. Actually, for any B e A(Fn),

(mi»f-E"f\"dP)'<mf'dP)'+ (pb^fpiP)

< 2( \B\f\VdP) VP.

(2.22)

Moreover, if (Fn}„>0 is regular and X < 0, then we can prove that Lp,X = Lp,X with equivalent norms (Theorem 3.1 and Remark 3.2).

Remark 2.6. By definition, if 0 > X > X', then we have that Li c LPrX c LPrX> with \\f ||L X, < \\f \Kx < \\f \\l„ and Li c Lprx c Lpx with \\f \Up, < \\f W^ < 2\\f H^ . If X < -1/p, then

L°p C Lp,x C Lp,x with \\f \\lp,x/2 < \\f \\Lpx < \\f \\Lp.

Remark 2.7. By definition and Remark 2.5, if Fo = (0, Q}, then Lpx c Lpx c L0p with \\f \\l <

\\f \lpx < 2\f \\LM.

Definition 2.8. Let BMO = L10 and Lipa = L1a if a> 0.

Our definitions of BMO and Lipa 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 p e [1, i) and X e (-i, i). For f e L1, let

llf= sup spkiw) i.,f ,dP) "'■ -=sup zipi? (m i.f - E-f ,pdP )"■

(2.23)

and define

Lp,x,f = f e L°p : 11/\\ w < œ}, lp,x,f = f e LP : \\fW^ < œ}• (Z24)

Note that the spaces Lp/x,f and LP/X/? can be defined without the assumption that every c-algebra Fn is generated by countable atoms.

Remark 2.10. By the definitions we have the relations Lp/x,f c LPrX with \\f \\LpX < \\f \1LpXF and Lprx,f c Lpx with \\f \\lp,x < \\f \\lp,x,f . If X > 0, then we can prove that Lprx,f = Lprx and Lp XFF = LP x with the same norms, respectively (see Proposition 3.8). If -1/p < X < 0, then LP/X,f ^ LP/X and Lp/x,f ^ LP/X in general (see Proposition 3.9).

Remark 2.11. It is known that, if {Fn}n>0 is regular and X > 0, then L1/X/? = LP/X/? with \\f \lixf < \\f \lpxf < Cp\\f \lixf for each p e [1, œ) (see, e.g., [8]). We also define weak Morrey spaces.

Definition 2.12. For p e [1, œ) and X e (-œ, œ), let

||f ^ = supsup^ SUp( Ml^ r (2.25)

P n>0 BeA(fn) P(B) t>0 \ 1 (B) /

for measurable functions f, and define

WLp x = ifeL0:| IflU <oo

{f e L0 : Wf \\WLp,x < ^• (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}„>0 be regular, Fo = (0, A} and (A, F, P) be nonatomic. Let p e [1, to). (i) If X < -1/p, then Lp, X = Lp X = Lp and

211 ^ ii Lp,x 11 Lp,x l

(ii) If -1/p <X< 0, then L0œ c LpX = LpX c L0p and

Lp 11J ii Lp,x 11 11l'

< c\\f \L

(iii) If X = 0, then L^ = Lp,0 c Lp/0 = BMO and

BMO < \\J II Lp,0 < Cplf II BMO'

(iv) If X > 0, then {0} = Lp,X c Lp,X = LipX and

LipX < \\f \\lp,x < CpW IILip."

Remark 3.2. We can prove (i) without the assumption that (Fn}n>0 is regular or that (A, F, P) is nonatomic. In (ii), we can prove that LP/X = LP/X and (1/2)||/H^ < \\f \\lm < C\\f Hl^

without the assumption that Fo = {0, A} or that (A, F, P) is nonatomic. To show L^ c Lp,X in (ii) and (iii), we can replace the condition that (A, F, P) is nonatomic by a weaker condition as in Proposition 3.6(ii), which follows from the condition that (A, F, P) is nonatomic. In (iv), we need the condition that (A, F,P) is nonatomic to show Lpx = {0}.

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

Lemma 3.3. Let {Fn}n>0 be regular. Then every sequence

B0 D Bi D ■■■ D Bn D ■■■ , Bn e A(Fn),

has the following property: for each n > 1,

Bn = Bn-i or (1 + r )P(Bn) < P(Bn_i) < RP(Bn),

where R is the constant in (2.21).

Remark 3.4. Since Bn e A(Fn) is an (Fn, P)-atom, we always interpret Bn-1 d Bn as the inclusion modulo null sets, that is, Bn-1 d Bn means P(Bn \ Bn-1) = 0. Therefore, Bn = Bn-1 means P(Bn \ Bn-i) = P(Bn-1 \ Bn) = 0.

Remark 3.5. By the lemma we see that there exists m such that Bm = Bn for all n > m, if and only if limn(Bn) > 0.

Proof of Lemma 3.3. Let Bn-t = {£n-i[jBn ] > 1/R}.Then B i. By the regularity we have

XBn < REn-l[XBn ]. This shows Bn-1 D Bn. In this case, Bn-1 D Bn-1 3 Bn, since Bn-1 e A(Fn-i). From the definition of Bn-1 it follows that Xb„^ < REn-1 [xb„ ]. Then we have

P(Bn-1) < P(Bn-1) = E[x~Bn_] < E[REn-1 [xb„]] = RE[xb„] = RP(Bn). (3.7)

Next we show Bn-1 = Bn or (1 + 1/R)P(Bn) < P(Bn-1). Suppose that

P(Bn-1) <( 1 + R)P(Bn). (3.8)

P(Bn-1 \ Bn) = P(Bn-1) - P(Bn) < P(Rn) < P(R-1)■ (3.9)

Therefore,

jt r 1 P(Bn-1 \ Bn) „ 1 /oin\

En-1 LXb„-1\b„J =-P(B ^— Xb„-1 < rXb„-1 ■ (3.10)

From the regularity and the inequality above it follows that

XB„-1\B„ < REn-1 [xb„-1\b„] < Xb„-1 ■ (3.11)

This means that Bn-1 = Bn. □

Proposition 3.6. Let {Fn}n>0 be regular, 1 < p < œ and 1 > -1/p.

(i) For a sequence B0 d B1 d ■■■ d Bk d ■■■, Bk e A(Fk ), let f0 = 0 and

fn = £ P (Bk /(P^f XBk - XBk-1), n > 1. (3.12)

Then f = (fn)n>0 is a martingale in M and in

(ii) Let 0 > 1> 1 > -1/p. If there exists a sequence B0 d B1 d ■■■ d Bk d ■■■, Bk e A(Fk ) and limk(Bk) = 0, then L0œ C LpA C Lpi. If F = {0, also, then L0œ g Lp,x g

Lp1 C Lp.

Proof. (i) By the definition of the sequence (fn)n>0, we have

a/ P (Bn-i)

En-1 [dnf] = P(Bn)^pp^B^En-i[xBn] -XBn-i) = 0,

(3.13)

for every n > 1. Hence, we obtain that (fn)n>0 is a martingale.

We next show that the sequence (fn)n>0 converges in Lp. If limk(Bk) > 0 then the convergence is clear by Remark 3.5. We assume that limk(Bk) = 0. Then, by Lemma 3.3, we can take a sequence of integers 0 = k0 <k1 < ■■■ <kj < ■■■ that satisfies

0+R) PB) < Kj < Mj

and Bkj-1 = Bk if kj-1 < k <kj .In this case we can write

fn = X P(Bk,)

x p\bkh

/ \ ~XBkj XBkj_ 1

P Bkj j j-1

Using (3.14) and the assumption X> -1/p, we have Z P(Bki) „(„ ) XBj - XBk-1

P Oku)

P (Bj)

< W^W^ + IK IIJ

x+1/p / 1 \-(X+1/p)j

ZP(Bk,)< 2^ 1 + R

P (Bn)X+1/p

Therefore, (fn)n>0 converges in Lp. We denote by f the limit of (fn)n>0. We can also deduce from (3.16) that

P(Bn) B

On the other hand, for B e A(Fn), we have

(f - Enf )xb =

— f If - Enf \pd^ < P(B )X

dP ) < P(Bn)X

f - Enf, (B = Bn),

v0, (B = Bn).

Combining (3.17) and (3.18), we have \\f H^^ < 1, that is, we get f e Lp;J

(3.14)

(3.15)

(3.16)

(3.17)

(3.18)

(ii) First we show L^ g LP,x g Lp ^. By Remark 2.6 we have L^ c LP/i c Lp ^. Then we need to show LP/0 \ L^ = 0 and LpA' \ Lvx = 0.

We consider fn in (3.12) for the sequence Bk, k = 0,1,.... Then we can write

t p (b^ ) \

fn = X I XBk,-XBkH I (3.19)

k <n\P{Bkj) ' 'J

we have

T^nf t^t Î \g - Ekj-1 g\"dP

}(BkH ) W hk,-1l 1 /

/ \1 / / w 1/p

P{Bk) (P{BkH \ Bk)

Pj\ P(Bk,-)

> R1 (1 + R)-1'-1/p^Bkj-)1 1 œ as j œ,

(3.20)

for 1 = 0 and we have f = (fn)n>0 e Lp,0. On the other hand, for a.e. w e Bkj \ Bkj+1,

f=j=fkj- xBkj=g( ^PB) - 0 -1 > R j+1} - 1

Then f /Li and Lp, 0 \ Li = 0. Next, let

_ ( xiY P (Bkj-1 ) \

*= (jfr) "x"kj-1 )■ (321)

Then g = (gn)n>0 e Lpi. On the other hand, since

\g - Ekj-1 g\ = P^)1' on Bkj-1 \ Bkj, (3.22)

——-7 f "T^tÎ \g - Ekj-1 g\*dP ,

P(Bj) \P{B*H) Bkj-1 1 y (3.23)

since 1' < 1. This shows Lp7 \ Lp,7 = 0.

Finally, if F0 = {0, also, then by Remark 2.7 and Lp^ \ Lp,1 / 0 we have Lp,1 g Lp. This shows the conclusion. □

Remark 3.7. In Proposition 3.6, f = (fn)n>0 in (3.12) converges in Lp as in the above proof. Moreover, the limit belongs to both LprX and LprX when -1/p < X < 0, since we will show that LpiX = Lpx in the proof of Theorem 3.1. However, it converges in neither Lp/X nor Lp,X. Actually, by a similar calculation to g = (gn)n>0 in the above proof, we have

I (f - fn) - EkH (f - fn)\ = \f - EkH f I = P (Bkl )X on Bkj-1 \ Bki, (3.24)

for n < kj-1, and then

(¿7 {j^-£*-1 /r

dP I > Rx(1 + R)-x-1/p. (3.25)

P{BkH) \nBkH) Bkj-i

By Remark 2.5, we have

2\\f - fn\\Lpl > \\f - fn\\Lp x > RX(1 + R)-X-1/p' (3.26)

This shows that (fn)n>0 converges in neither LpxX nor Lp,X. Proposition 3.8. Let 1 < p < œ.

(i) If F0 = {0, A} and X <-1/p, then L°p = Lp,X = Lpxf = Lp xx = LV,X,f with

^ .....^ .....LP < \\/IU, < \\/Lp,,< 2\\/ \\lpX, ■ (3.27)

(ii) If X > 0, then Lp,x = Lv,x,f c Lp,x = Lp,x,F c Lp with

Lp U J U LpX? " ^ " Lpr1 ' ' J ' ' LpX9 uj 11 Lp,1

(3.28)

Proof. (i) Let 1 <-1/p. For any / e LLp and any B eFn

pw(pb)îB 1/|PdP = P (b)-1-1/p(Ib 1/h1/p

< P(0)-1-1/p( Jj/\PdP) Vp = W/IILp (3.29) = P(Q)-1-1/p(£ |/ - Eo/\PdPj1

Then by the definition of the norms and the assumption F0 = (0, A} we see that \\f \\Lpx F = \\f \\v = \\f \L < \\fw < \\f \k i F • Bythe same observation as Remark 2.5 we have \\f \lp xF <

2\\f \\Lp/i/f •

(ii) Let X > 0. By Remark 2.10 we need to show only LP/X c Lpx,f with \\f \\Lp x F < \\f \\Lp x and Lpx c LpX/f c L0p with \\f \\lp < \\f U^ < \\f W•

Let f e Lp/X. For any B e Fn, there exists a sequence of atoms Be e A(Fn), £ = 1,2,..., such that B = ueBe and P(B) = P(Be). Then

f If \PdF = E f If \PdP < EP(Be)AP+1H/11^ < P(Be)V+1H/11^' (3.30)

J B / J Be /

since Xp + 1 > 1. Therefore f e LpXf and 11/llw - 11/• Similarly, we have \\f \\lpX9 -11/|lp,x for f e Lp,x. By the definition of LpXf norm we have \\f ^ = \\f - EofIlLp -P(Q)X+1/p\\f \lpXF = \\f \lpXF for f e LpXf. □

Proof of Theorem 3.1. (i) We have the conclusion by Proposition 3.8 without the assumption that (Fn}„>0 is regular or that (Q, F, P) is nonatomic.

(ii) By Proposition 3.6 we only need to prove LPx = LP/X with (1/2)\\f \\LpÀ - \\f \\LpX -C\\f \\LpÀ. The first norm inequality follows from Remark 2.5. We show the second one. Note that we do not need the assumption that F0 = (0, Q} or that (Q, F, P) is nonatomic. Let f e Lpx. Then, for any B e A(Fn),

Îb 1 f 1 pdP) 1/P < ( P(B) i I f - Ef «pdP) VP+\ P(B) if (-)dP

< P(B)

\ f (w)dP

(3.31)

f f (w)dP

(3.32)

If B e F0, then Enf = Oon B. Assume that B / F0. By Lemma 3.3 we can choose Bkj e A(Fkj), 0 = k0 <ki < ••• <km < n, such that Bko d Bkl d Bkl d---d Bkm = B and that (1 + 1/R)P (Bkj) < P(BkH) < RP(Bk). Then, since

P (Bko )

- f f (w)dP = 0,

:o) •'Bk,

(3.33)

we have

mi/(M)dP - m Lf (M)dP - P(kr\j(w)dP

1 i) \jwai'=vm Vw^-Pü-)

m ' i r „ . _ 1

- E ( -T^T Î f {M)dP - "T^T f f (w)dP j-\Pfa))*, P{BkH) Jj

-£ -7^) [f - Ekj-if] (w)dP.

By Holder's inequality and the assumption 1 < 0 we have

[ f (^)dP < E ( [ If - Eh- f \PdP

1 j . If - ^ rdP N

j-a P{Bki_j ^

EP(Bk,-i)-

1 v m-j+1

< X^1 + r) P (Bkm )j ......

P(B) N, „lpy

(3.34)

(3.35)

Therefore we have ||f ||L 1 < \\f ||l a.

(iii) Let 1 = 0. By Remark 2.6 and Proposition 3.6 we have Li c Lp,0 with ||f ||L < ||f L and Li C Lp,0.

Next we show Lpr0 c Li and ||f ||Li < ||f H^.

Let f e Lp0 and f = 0 a.e. Take a positive number r such that P(|f | > r) > 0. For any e > 0, there exists n and B eFn such that

P (B n {\f \>r}) > (1 - e)P (B) , (3.36)

because F is generated by UnFn. For the above B, we can take a sequence of atoms Be e A(Fn), £ = 1,2, ■■., such that B = U£B£ and P(B) = ^£P(B£). Hence, by the pigeonhole principle, there exists B' e A(Fn) such that

P (B'n {\f\>r}) > (1 - e)P (B) (3.37)

Journal of Function Spaces and Applications Therefore, we have

if to * pb) L fiPdP

* HbO f If lPdP

1 (B) JB'n{\f\>r} (3.38)

^P(B'n {|fI >r})rp - P(B')

* (1 - e)rp.

This shows that P(|f | > r) > 0 implies ||f ||Lp0 * r. Then we have the conclusion.

By Proposition 3.8 and Remark 2.11 we have that BMO (= L1 r\) = LPr\ with equivalent

norms.

(iv) Let X> 0. For f e Lpx we suppose that there exists r > 0 such that P(|f | > r) > 0. Then, for any e > 0, we have the same estimate as (3.37). Moreover, we can decompose B in (3.37) to atoms in A(F«+1) and we have the same estimate as (3.37) for some atom in A(F«+1). Therefore, we can take an atom B' in A(Fn+m) for large enough m such that B' satisfies (3.37) and P(B') < e. Hence we have

» > 1 , f if\pdp > (l^r! > (1 " f)rP

lp,X > \i+ApJB, fl dP > \Xp > XP ' (3-39)

,(B<)1+XP JB' P(&)

that is, r < eV(1 - e)i/p\\f \\Lpx. This contradicts that r> 0. Then f = 0 a.e. and LP/X = {0}. By Proposition 3.6 we have that {0} g LPr\.

Finally, by Proposition 3.8 and Remark 2.11 we have that L1 x = Lpx with equivalent norms. □

Next we prove that Lpxf g Lpx and Lpx,f g Lpx in general by an example.

Proposition 3.9. Let (Q, F, P) be as follows:

Q = [0,1), A(Fn) = {In,j = [j2-n, (j + 1)2-n) : j = 0,1.....2n - 1}, (3.40)

Fn = a(A(Fn)) , F = ^[JFn^, p = the Lebesgue measure. (3.41)

If -1/p < X< 0, then LPri,F g LPri and LP xff g LP,> Proof. We construct f such that

f e LP/1 \ lP/1/f, f e LP/1 \ £pxf- (3.42)

Step 1. Denote the characteristic function of In/j by xn,j and let

fn = E fn+m,2mj * fn,j = P(In,j) (Xn+1,2j — Xn+1,2j+l) , (3.43)

where we choose m such that

P(In+m,o)pX+1 < P(Infi)- (3.44)

Note that Inj = In+1,2j U In+1,2j+1 and \fn,j| = P(Infi)XXnj.

If k < n + m, then Ekfn = 0, and \fn - Ekfn\ = \fn\ = P(In+m/0)X E^1 Xn+m,2mj. Hence

P (Ik/)

n — Ek fn I dP

fn — Ekf

•'It ,

pl l №p) (345)

T>t r ^ / ^jw», ciW P (W2"/)

= P (ln+m0) ( -P(M-

If k < n, then the number of the elements of (j : In+m/2mj c Ik,e} is the same as of (j : In/j c Ik,e}. Hence

1 'lit, fn — Ekfn|Pd^ ^ L, ^ VP

P(Ik,) Jh, / Jh, (3.46)

< P(In+mfi)^P(^) P < 1 < P(Ik,)\

\ P (In,0) /

where we use (3.44) and X < 0 for the last two inequalities. If n <k < n + m, then the number of the elements of (j : In+m/2r«j c Ik,e} is one at most. Hence

1 f If.- f*dpf ^^ f \f„\^

P(Ik'e)j It,, /

< P(In+m,0)^PP^) < P(Ik,)'

where we use -1/q < X < 0 for the last inequality. In the above, if Ik,e = In+m/0, then the equality holds.

If k> n + m, then fn - Ekfn = 0 and

1 . \ 1/p

1 I I - I'^D 1 ^ TUT „\a s TUT, „\A

P (Iks)

IfnIPdP ) < P(In+m,0)A < P(Ikfi)1. (3.48)

J Ik f /

Therefore,

fn e nLp> iifnl^^ = Ifn^Lp, = 1. (3.49)

On the other hand, for the set B = U2=01 In+m,2mj e Fn+m,

(3.50)

(p(WJ E f- -( m i. f'dP)

= P(In+m,

= (2-nP (B))A.

Therefore, yfnyL'iXiF' ||fn|L',i,? * 2-nA ^ ^ as n ^ TO.

Step 2. Let fn be as in Step 1. If k < n, then, by the same observation as Step 1 we also have that

fnXk,f e L',A nL'A ifnXk,f 11 L'ry = 11 fnXk,f1 L',x = 1 (3.51)

Moreover, if k < n, then, for the set B = (U;=o In+m,2mj) n Ik,f e F

1 f \ ' / 1 f \ '

p. J | fnXk,f - En+m [fnXkA IP = p(B) J . | fnXk,f T dPj

= P (In+m,0)A (3.52)

(2-n+kP (B)) A.

Therefore, ||fnXk/ hl^f, llfnXkf lu^ * 2(-n+k)A. Step 3. Let fn be as in Step 1 and let

f = 2 2nA/2f2nXn,1. (3.53)

Note that X^ 1 = 1. Then

<» 2X/2

Ilf Hl^ IIf Ikx <X2nX/2 = T^. (3.54)

On the other hand, by the same observation as Step 2, we have that

> 2nX/2 x 2(-2n+n)X = 2-nX/2 (3 55)

LpXF ii-' "lp/X/F

for all n. This shows (3.42). □

At the end of this section we prove the relation of \\f \\L1X and \\f\\WLp. Recall that

WLp = suP tP(\f\>t)1/P- (3.56)

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

Proposition 3.10. If 1 < q <p < to and -1 /p = X, then

Hf H L1 / X < I|f HLq,X < CHf H (3.57)

Proof. The first inequality followed from Holder's inequality. To show the second inequality, we assume that \\f \\WLp = 1 and prove that \\f \\LqX < C. From \\f \\WLp = 1, we have P(\f \ > t) < t-p. For any atom B e A(Fn), let n = P(B)X = P(B)-1/p and

(f (w) , \f (w)\>n f = fn + fn/ fn(w) = fn ! (3.58)

10/ \f (w)\ < n

L ]fn(w)]'dP - pbxtJ0 ^ f 1 >^0dt

P(B)qX+1 J^^' ' " P(B)

- P(B)qX+\jo - ' Jn

c n c °°

J p ( I fn I >n) (qt7-1 ) dt TO i-P (qt7-1 ) dt

<—i-^ + -^\q-p = P

p (B)qX+^ p - q/ p - q'

Since \fn(w)\ < n we have

(3.59)

P(5)^(pk l f w)- PB)'

dP) -—l—r = 1. (3.60)

We get the conclusion.

4. Maximal Function

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

iifllL' < '-i iif llv f e L' ('> 1), (4.1)

llflU <llf llL1- f e L1. (4.2)

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 {Fn}n*0.

Theorem 4.1. Let 1 < ' < to and -1/' < A < 0. Then, for f e Li,

llfllL', < C'Ufll^, ifP > 1

llflU, < C1lf ||l^, fp =1.

Proof. Case 1 (' > 1). For any B e A(Fm) and m * 0, let f = g + h and g = fx.. Then, using (4.1), we have

f (g*YdP < f (g*)PdP < f IgI'dP = f IfI'dP. (4.4)

JB JQ JQ JB

1 (jb L (g*)'d^1/P - ifwl*. (4.5)

P(B)A P(B) B

Next, take Bn e A(Fn), n = 0,1,...,m, such that B = Bm c Bm-1 c ... c B0. Then, for a.e. w e B,

Enh(w) = -

0, (n * m),

hdP, (n<m).

, _ , , (4.6)

P.), Bn

If n <m, then

|Enh(w)l < ( jB) {. Ihl'd^ " < P(Bn)A\\f Hl'A < P(B)A\\f Ula (4.7)

since A < 0. Hence

h* < P(B)A||f on B. (4.8)

L{hrdP) < llf 'v <49)

By (4.5), (4.9), and the inequality f * < g* + h*, we have

(pb is f)vdpy - if(4i°)

P(B)X\P(B) Jb

which shows the conclusion.

Case 2 (p = 1). Let g and h be as in Case 1. Using (4.2), we have, for all t> 0,

tP(B n g > t}) < tP(g* > t) < f |g|dP TO \f\dP (4-11)

< If IL,- (4-12)

and then

1 tP(B n {g* >t})

P(B)A P(B) " llLu

We also have (4.8) for the case p = 1. Then

1 tP(B n{h* >t}) .. PBf P(B) < llf (413)

Therefore we have the conclusion. □

5. Fractional Integrals

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 Ia as a martingale transform.

For normed spaces M1 and M2 of functions, we denote by B(M1, M2) the set of all bounded martingale transforms from M1 to M2, that is, Ia e B(M1f M2) means that

suPll (Iaf ) Jm2 < C suP|f"!M1' (5.1)

for all Mi-bounded martingales f = (fn)n>0.

We state our results in Section 5.1. To prove Theorems 5.1 and 5.5 we show the pointwise estimate for Iaf with the assumptions \\f ||lp < to and \\f \\lpA < to, respectively. To avoid repetition we prove first Theorem 5.5 in Section 5.2 and then Theorem 5.1 in Section 5.3. The proof of Theorem 5.8 is in Section 5.4.

5.1. Boundedness of Fractional Integrals

The following is for L'-Lq boundedness.

Theorem 5.1. Assume that {Fn}n*o is regular. Let 1 < ' < q < to, -1/' + a = -1/q. Then, for f e L1,

Hf ULq < CUf llV if '> 1r (5.2)

ll (Iaf )1lwLq < cII/IIli' f' = 1. (5.3)

Remark 5.2. Let f = (fn)n*0 be an Li-bounded martingale. For each Li-function fm, m * 0, consider the corresponding martingale fm = (En[fm])n*0 = (fmin(n,m))n*0. Then (Iafm)n = (Iaf )n for m * n. Therefore, from (5.3) it follows that, for m * n,

ll (Iaf )nllwLq = ll (Iafm^nllwLq - ll/»llL! < sup||fn||L1. (5.4)

q q n>0 v '

This shows that

sup \ (Iaf) n \ WLq < C sup \\ fn \ L1. (5.5)

n 0 q n 0 1

Remark 5.3. Let a martingale f = (fn)n*0 be Li-bounded. Since WLq c Lq1 and ||(Iaf )n|Lfl < ll(Iaf )n|WLq for 1 < qi < q, from Remark 5.2 it follows that the martingale ((Iaf )n)n*0 is Lq1-bounded and that it converges in Lqi. Denote this limit by Iaf. Then, Iaf = ^to=0 1 dkf and En[Iaf ] = (Iaf )n.

If' > 1, then the same observation as in Remark 5.2 with (5.2) shows that

sup\\ (Iaf )n|Lq < C sup\\ fn \L'. (5.6)

n*0 q n*0 ' V '

Hence we have the following corollary.

Corollary 5.4. Assume that {Fn}n*0 is regular. Let 1 < ' < q < to, -1/' + a = -1/q. Then

Ia e B{L',Lq), if '> 1, Ia e B(Li, WLq), if ' = 1.

For Morrey norms, one has the following.

Theorem 5.5. Assume that {Fn}n*0 is regular. Let 1 < ' < to, -1/' < A < 0, and ¡i = A + a < 0. Then, for f e L1,

WafTH^ <C11f11L'rA' if'> 11 <q< (/)' (5.8)

H(Iaf)1|Lq, < CHfHLly if' = 1 1 < q< (A/i)'* (5.9)

ll (Iaf )1wLw < CHf Hlu' f ' = 1 q = (A/i)'. (5.10)

Note that Theorem 5.1 is not a corollary of Theorem 5.5,since ||f ||L' —/||f |L',-1/' < ||f ||L' in general.

Remark 5.6. In order to prove (5.8) it suffices to prove it in the case where q = (A/i)', since ||f |Lfl,i < ||f 11Lq,A for q1 < q by Holder's inequality. The inequality (5.9) follows from (5.10),

since Hf |Lq1,i < ||f ^WLqA for qi < q.

By the same observation as in Remark 5.2 we have the following.

Corollary 5.7. Assume that {Fn}n*0 is regular. Let 1 < ' < to, -1/' < A < 0, and ¡i = A + a < 0. Then

Ia e B(L'rA, Lq,i) , if '> 1, 1 < q < {A/i)',

Ia e B(Li,A, Lqi), if' = 1, 1 < q< (A/i)', (5.11)

Ia e B(Li,A, WLqi), if ' = 1, q = (A/i)'.

For Campanato spaces, one has the following.

Theorem 5.8. Assume that {Fn}n*0 is regular. Let 1 < ' < to, A * -1/', 1 < q < to, and ¡i = A + a> 0. Then

paf < CHf lUy f e L1' (5.12)

Ia e B(L'aA, Lqi), (5.13)

where Iaf in (5.12) denotes the limit function described in Remark 5.3, for f = (Enf )n*0, f e L1.

Remark 5.9. If ¡i < 0, then, by Theorem 3.1 (ii), the inequalities (5.8) and (5.9) in Theorem 5.5 hold with L',A and Lqi replaced by L',A and Lqi, respectively.

If 1 <'< to and -1/' = A, then ||f ||Ai ~ Hf ||lia < Hf Hwl, < Hf by Remark 2.5 and Proposition 3.10. Since L' g L1 A in general, the following is an improvement of the results in [21].

Corollary 5.10. Assume that (Fn}„>0 is regular. Then, for f e L1,

II«fIIbmo < CIIfHl1^1/p, if - p + a = 0,1 <P< to,

II«f Lp, < CIIf II w if - p + a = ,> 01 <P< to, (5.14)

IMIlip« < CIIf IIвMO,

IIVIILipr < CIIf IIL1p/ if , + a = Y,> o

I« e B(L\r-1/p,BMO), if - p + a = 0, 1 <p< to,

1«eHu^pup«^ if-- +« = ,>a 1 <p<a^

p (5.15)

I« e B(BMO,Lip«),

I« e B^Lip,,LipY), if , + a = Y, ,> 0.

5.2. Proof of Theorem 5.5

By Remark 5.6 we only need to prove (5.8) and (5.10) in the case q = (X/p)p. So we always assume that q = (X/p)p in this proof.

First we show the pointwise estimate

(Iaf)» < Cr^UfU-L«/1, (5.16)

where C is a positive constant independent of w. To do this we prove that, for any m > 1 and any Bm e A(F

| (fmM < C/WA||/\\Z\ V e Bm. (5.17)

Take Bk e A(Fk), 0 < k<m, such that Bm c Bm-1 c ... c B0 and let

K = {k : 0 <k < m,Bk = Bk-i} = {ki,k2,...,k}, (5.18) where 0 = k0 <k1 <k2 < ... <ke. Then, by Lemma 3.3,

1 + -1) bk, < bk-i < Pbk, on Bm. (5.19)

Hence, we can write

(fm = X KH dk]f = £ Kh dk]f on B

0<kj <m ]=1

(5.20)

since bk = bk-1 and dkf = 0 for k / K. Note that for w e Bm

\dk,f (w)\ = f (w) - fk]-i (w)\ < f (w)\ + |fk]_i

-rrf fdP

P(Bk]) JBk,

f \f\'

P Bk]-1 Bk]-1 i/p

]] < ( P(Bk>y + P(ßkhi^ 1

P(B]i) Ki

( \f\'

(5.2i)

<( i + ( r) lbk]-i (w)\\JWU

Then, for w e Bm and for 0 < n < m,

^bk-iMadkf (w)

X bk]-i(w)ifiL

i + (i/R)1

i - (i + i/Rf

bn(wYllf I

(5.22)

On the other hand, for 0 < n < m, letting j(n) = min{j : n < kj}, we have, for w e Bm

^ bk-i (w)adkf (w) = ^ bk]-i (w)adk]f (w)

k=n+i ] =](n)

= X bk]-i (wf (w) bk]-i (w)afk]-i (w) ]=](n) ]=](n)

(5.23)

bk£-i (w)f(w) + £ (bk]-i (w)a - bk] (w)^fk] (w)

]=](n)

- bk](n)-i (w)afk](n)-i (w)

0<k]<n

< bke-1 (v)a/*(v)

^ bk-1(v)adk/(v)

Here, let

+ X | bkj-1 (v)a - bk] (v)^/* (v)

j=j(n)

+ bk](n)-1 (V)a/*(v)

< 2bk](„)-1 (v)a/*(v) = 2bn(v)a/"(v).

(5.24)

( r(v)\

< b0(v)

A2 = A \ A1

(5.25)

If v e A1 n Bm and

/ *(v)

< bm(v),

(5.26)

then, by (5.22) we have

K/m(v)| < Mv)l/|L <

/ *(v)

mw \\J II r , < I ll ,-11 I 11/ II L

Lr,X 1 11 4- M I 11 11 L

rX " V lUII. / "j "lp-1

= rw^n/nz1, (5.27)

since p < 0 and 1 - p/X = -a/X. If v e A1 n Bm and

bm(v) <

/ *(v)

(5.28)

choosing n such that

R-1bn(v) < ( / (.v) ) < bn(v),

(5.29)

we have by (5.22) and (5.24)

£ bn(wf II/||v + bn(w)ar(w)

, V ^/1 , V a/1

< ) ||/|lv + (n^L) /(„) (5.30)

Lp,1 / \ II II Lp,1

< /^Г1/IIZ1-

If w e n Bm, then by (5.24) we have

К !«/)m(w)l =

Yjbk-\(w)adk/(w)

bo(w)a/*(w)

/ \ a/1

f f * (wM г*/, л _ r*/. и -a/1

(5.31)

< ( itF-) f(w) = rwrll/II

Therefore, we have (5.17).

Next, applying the boundedness of the maximal function (Theorem 4.1), we show

(p. I. '('af(w))']qd^< P(BTVlit,, ¡f '> 1- (5.32)

sup ,(P (B " 'P^' >"))1/q < p^l., if' = 1, (5.33)

for any n and any B e A(Fn)• To show (5.32), using (5.16) and the relations q^/X = p and -a/X = 1 - p/q, we have

PB) i^1*/(w» 1"dP) <( m \j-(w)q"'1dP) II/К/

J /*(w)pdP

1 r \ (1/p)(p/q) , ,

PB |B/ '(w)pd^ I/IC"

(5.34)

Applying the boundedness of the maximal function, we have

(i/p)(p/q)

/ 1 Г \ (1/p)(p/q) / \ p/q ,

КШ JB/*(w)PdPj < (P(b/H/ilJ ? < P(ВЩ/f^. (5.35)

Then we have (5.32). To show (5.33), using (5.16) and the relations p = 1, p/X = 1/q and -a/X = 1 - 1/q, we have

/ P (B H {(Iaf )' >t})\1/q < fP(B ^ {(/^llfCM) X 1/q

sup \—m—; < supf 1-m-y

( P (B h {(f *)1/q f \\H/q> f})^ = sUP f 1 P(B) '

'p (b n {(r)1/q>t}y 1/q

t>0 V P(B)

1—1/q

= suP f 1 p (B)-^ J

P (B n {(f *) >t})

= ( sUoP ' P(B)

1—1/q L1,X '

(5.36)

Applying the boundedness of the maximal function, we have

sup tP(B n !(/■) >>)) ^ < (p^,/.^y/q s p(B)1,(537)

Then we have (5.33). The proof is complete.

5.3. Proof of Theorem 5.1

Let X = -1/p and p = -1/q. Then p/X = p/q and -a/X = (1 /p - 1/q)p = 1 - p/q. In this case the pointwise estimate (5.16) implies

(.Iaf)» < Cf (V)p/q\\f IlL-pP/p < C/*(^)p/q||/||L^p/q. (5.38)

Applying the boundedness of the maximal function on Lp, we get the conclusion.

5.4. Proof of Theorem 5.8

Note that, for f e L1, ||f |Up,X = ||f - Eof IUp,X and Iaf = Ia(f - Eof). Then it is sufficient to prove (5.12) only for f e L1. Therefore, (5.12) follows from (5.13) and Proposition 2.2(ii).

By Holder's inequality we have ||f ||l1X < ||f ||lpX for p > 1. Furthermore, by

Theorem 3.1 (iii) and (iv) we have ||Iaf ||l ~ ||Iaf ||l1 for q > 1. So we only need to prove that 1

sup\\ (Iaf )m\l1p < C sup \ \ fm \ l1 x' (5.39)

m>0 m>0

for all Li/X-bounded martingales f = (fm)m>0.

To prove (5.39) we show that, for any n, any m> n and any Bn e A(Fn),

I" I (Iaf )m — (Iaf)n\dp < P(Bn)1+P\\fm\\Al-

(5.40)

Note that

(fm - (fn = X K-Af = X (bh - ba)(fk - fn) + K-f - fn). (5.41)

k=n+1 k=n+1

Since bm is nonincreasing with respect to m, we have

bam-1\fm - fn\dP < P (Bn)TO |fm - fn\ dP < P(Bn)1+P\\fm. (5.42)

J Bn J Bn

Therefore, to prove (5.40), we only need to show that

X b-1 - K) \fk - fn\dP < P(Bn)1+P\fm\lu Vm>n- (5.43)

k=n+1^ Bn

Using the inequality |fk - fn |< Ek |fm - fn| for n <k < m, we have [ (bt-1 - bt)\fk - fn\dP TO (bt-1 - bt)Ek \fm - fn\dP TO (bt-1 - bt)\fm - fn\dP.

Bn Bn Bn

(5.44)

Therefore, we have

m f m f

X (bt-1 - ^\fk - fn\dP < £ (bt-1 - bD\fm - fn\dP k=n+1^Bn k=n+1^Bn

TO (bn - bm)\fm - fn\dP

< P(Bn)TO \fm - fn\dP

(5.45)

< P(Bn)1+P\\f„

We have obtained (5.43). The proof of Theorem 5.8 is completed. Acknowledgments

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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