Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 690258,14 pages http://dx.doi.org/10.1155/2013/690258

Research Article

Martingale Morrey-Hardy and Campanato-Hardy Spaces

Eiichi Nakai,1 Gaku Sadasue,2 and Yoshihiro Sawano3

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

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

3 Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Osawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan

Correspondence should be addressed to Gaku Sadasue; sadasue@cc.osaka-kyoiku.ac.jp

Received 10 June 2013; Accepted 17 August 2013

Academic Editor: Natasha Samko

Copyright © 2013 Eiichi Nakai et al. 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 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.

1. Introduction

Lebesgue spaces Lp and Hardy spaces Hp play an important role in martingale theory and in harmonic analysis as well. Morrey-Campanato spaces are very useful to know more precise properties of functions and martingales. It is known that Morrey-Campanato spaces contain Lp, BMO, and Lipa as special cases; see, for example, [1, 2].

In martingale theory, Weisz [3] introduced martingale Lipschitz spaces for general filtrations [Fn}n>0 andprovedthe duality between martingale Hardy spaces and martingale Lipschitz spaces. This result was extended to generalized martingale Campanato spaces and martingale Orlicz-Hardy spaces in [4]. Recently, martingale Morrey-Campanato spaces were introduced in [5], where each sub-u-algebra Fn is generated by countable atoms.

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 [3-5]. To do this, we first introduce generalized martingale Morrey-Campanato spaces by using subfamilies |B„}„>0 of the filtration |FJ„a0 with Bn c Fn for each n> O.We establish Burkholder-type equivalence and discuss equivalence between martingale Morrey spaces and martingale Cam-panato spaces in a suitable condition. We also establish a John-Nirenberg-type theorem for generalized martingale Campanato-Hardy spaces; see Theorem 15.

On these martingale spaces, we introduce generalized fractional integrals as martingale transforms and prove their boundedness. Our result extends several results in [5-7] to these spaces. The fractional integrals are very useful tools to analyse function spaces in harmonic analysis. Actually, on the Euclidean space, Hardy and Littlewood [8, 9] and Sobolev

[10] investigated the fractional integrals to establish the theory of Lebesgue spaces and Lipschitz spaces. Stein and Weiss

[11], Taibleson and Weiss [12], and Krantz [13] also investigated the fractional integrals to establish the theory of Hardy spaces; see also [14]. The Lp-L boundedness of the fractional integrals is well known as the Hardy-Littlewood-Sobolev theorem derived from [8-10]. This boundedness has been extended to Morrey-Campanato spaces by Peetre [1]and Adams [15]; see also Chiarenza and Frasca [16]. In martingale theory, based on the result on the Walsh multiplier by Watari [7, Theorem 1.1], Chao and Ombe [6] proved the boundedness of the fractional integrals for Hp, L p ,BMO, and Lipschitz spaces of the dyadic martingale. The boundedness of the fractional integrals for martingale Morrey-Campanato spaces was established in [5]. For other types of operators for martingales, see the recent workby Tanaka and Terasawa [17].

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, are dependent on the subscripts. If f < Cg, we then write f < g or g > f; and if f < g < /,we then write f ~ g.

2. Definitions and Notation

Let (Q, Z, P) be a probability space and F = {Fn}n>0 a nondecreasing sequence of sub-a-algebras of Z such that Z = <j(\Jn Fn). For the sake of simplicity, let F-1 = F0. The set B e Fn is called atom, more precisely (Fn, P)-atom, if any Ac B, A e Fn, satisfying 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. It is known from the Doob theorem that if p e (1,ot), then any L^-bounded martingale converges in Lp. Moreover, if p e [1,ot), then, for any f e Lp, its corresponding martingale (fn)n>0 with fn = Enf is an L^-bounded martingale and converges to f in Lp (see, e.g., [18]). For this reason a function f e L1 and the corresponding martingale (fn)n>0 will be denoted by the same symbol f.

Let M be the set of all martingales such that f0 = 0. For p e[l, ot], let L0p be the set of all f e Lp such that E0f = 0. For any f e L0p, its corresponding martingale (fn)n>0 with fn = Enf is an Lp-bounded martingale in M. For this reason, we regard L0p as a subset of M.

Let B = {Bn}n>0 be subfamilies of F = {Fn}n>0 with Bn c Fn for each n> 0. We denote by B c F this relation of B and F.

In this paper, we always postulate the following condition on B

There exists a countable subset B0 c B0

such that P( J ß) = 1

We first define generalized martingale Morrey-Campanato spaces with respect to B as follows.

Definition 1. Let B c F, p e [1,ot), and 0 : (0, l] ^ (0,ot). For f e L1, let

= sup sup

n-/oBJn$(P(B))\P(B) iB

f \f\pdp B

sn^PBseuP„ Jim(W)llf-EnfUP)lP'

= su su 1 ( 1

$(P(B))(P(B) Jb

f \f-En-J\pdP B

and define

-p4 ^[./ - ■ wJ wL^ Lp4 = Lp4 (B) = [f e L0p : \\f\\ < œ} , (3)

Lp, = Lp4 (B) = [feL0n : ll/ll, , <œ},

= L-p4 m = \feL<

Remark 2. By the condition (1), the functionals

IIare norms on L „.

Remark 3. Let f e L0V Then, f e ^ if and only if its corresponding martingale (fn)n>0 is Lp^-bounded; that is, supn>oWf„WL < œ. The same conclusion holds for Furthermore, if each sub-a-algebra Fn is generated by countable atoms, B = [A(Fn)]n>0 and 0 is almost decreasing, then the same conclusion holds for Lp More precisely, see Proposition 8.

Remark 4. In general, WfWL ^ < 2W/WL and hence Lp4 c

>0. Actually for any B e Bn,

(f„\f-*nf\'lr)"' <(J,\f№)"'

f Kf\pdP

f \f\pQ B

Similarly, WfW^ < nf\\3-4 and L^ c L p4.

Definition 5. For ^ = l, denote Lp4 and Lp^ by BMOp and BMOp, respectively. For <p(r) = ra, a > 0, denote Lp4 and

LP,t by Lipp,a and ^p-,^ respectively.

If $(r) = rx, \ e (-ot, ot), then we simply denote Lp4, Lp>0, and Lp4 by LpM LpM and respectively.

A function <p : (0, l] ^ (0,ot) is said to be almost increasing (resp., almost decreasing) if there exists a positive constant C such that

0 (s) < (t) (resp., 0 (t) < (s)) for 0< s<t <l.

For the case B = F, the spaces BMOp(F) and Lip (F) with a > 0, were introduced by Weisz [3].

Recall that A(Fn) is the set of all atoms in Fn and let A = {A(Fn)}n>0. Suppose that each sub-a-algebra Fn is generated by countable atoms for the time being. Then, BMOp (F) = BMOp(A) and Lip pa(F) = Lippa(A); see [5]. In general, if 0 is almost increasing, then

Lp4 (F)=Lp4 (A),

Lp4 (F) = Lp4 (A)

LPp4 (F)= LPp4 (A),

with equivalent norms, respectively. However, if 0 is not almost increasing, then these equalities fail in general; see [5].

In this paper, we do not always assume that each sub-a-algebra Fn is generated by countable atoms. Let

A(Fn)± = |B e Fn :P(BnA) = 0VAeA (Fn)}, (7) and let

Bn = A(Fn)uA(Fn)± (n> 0). (8)

In this case, if 0 is almost increasing, then we will show that Lp4 (F) = Lp4 (B),

Lp4 (F) = Le é (B)

(F) = Lp4 (B),

with equivalent norms, respectively (see Proposition 9). Moreover, if F0 is nonatomic, then Bn = Fn for all n > 0. If each sub-a-algebra Fn is generated by countable atoms, then Bn = A(Fn) for all n > 0. Therefore, our definition generalizes those in [3-5].

Next we define Morrey-Hardy and Campanato-Hardy spaces, based on square functions, with respect to B as follows. For f e M, we denote by Sn(f) and S(f) the square function of f:

Sn (f) = (l\dj\2)' , S(f) = (z\dkf\'

\k=0 ) \k=0

where dkf = fk - fk-1 (n > 0, with convention d0f = 0 and S_1(f) = 0). We further define

s(n) (f) = (s(f)2 - sn(f)2f = ( X M2 ) . (11)

Definition 6. Let B c F, p e (0,œ), and 0 : (0,1] (0,œ).Forf=(fn)nx> e M,let

= su su 1 ( 1

\\hsp4(b)

= sup sup

nloBB $(P(B))\P(B)

i \1/p jB S(f)pdP) ,

S(n)(f)pdP) ,

= sup sup

n>0 BeB„ $(P(B))\P

1/p (12)

and define

_rS < œ

4 = Hsp4 (B) = {fe M : Hsp4 = Hsp4 (B) = {fe M : \\f\\HU < œ} , (13) HSp-4 = Hp-4 (B) = {fe M :

,s- < œ

By (1), the functionals quasinorms on M.

\\HS , W/Whs , and W/Whs- are

Remark 7. If we take ^ = 1 and B = F, then the norm

coincides with the norm

s in [19, Definition

2.45]. In this point, our notation is different from the one in [19].

In the end of this section, we present the definition of regularity on F and the doubling condition on 0. The filtration F = {Fn}ni0 is said to be regular, if there exists a constant R>2 such that

fn < Rfn-1

holds for all nonnegative martingales (fn)n>0. We say the smallest constant R satisfying (14) the regularity constant of F. A function 0 : (0,1] ^ (0,rn) is said to satisfy the doubling condition if there exists a positive constant C^ such that

<ï(s) t(t)

< Cé Vs, t e (0,1] with — <- <2. (15)

The smallest constant C^ satisfying (15) is called the doubling constant of 0.

3. Properties of Morrey-Hardy and Campanato-Hardy Spaces

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 6.

First we state basic properties of the norms.

Proposition 8. Let B c F, p e [1,ot) and $ : (0,1] ^ (0,rn). Let f e L1 and let (fn)n>0 be its corresponding martingale; fn = Enf. Then

IL < snuPWfnK

\\*„ = sup\\fn\\Lp4

= sup||/n||

Moreover, if each sub-a-algebra Fn is generated by countable atoms, B = A, and $ is almost decreasing; that is, there exists a positive constant C0, such that^(t) < C0$(s) for 0 < s < t < 1, then

< sup||/n|

^ < C°H

Proposition 9. Let A(Fn) U A(Fn)± c Bn c Fn(n > 0). If 0 is almost increasing; that is, there exists a positive constant C0, such that^(s) < C0$(t) for 0 < s < t < l, then

wlp4(b) < Wfh^f) < Co\\f\^Lp4(b)'

and the same conclusions hold for w • w l , W • 11ll- , W • W1-7S ,

J L p, $ p, $

W • Whs , and W • Whs- . Consequently

Lp4 (F) = L oA (B), L„a (F) = L„a (B),

LpA (F) = LpA (B), HpA (F) = HpA (B), (19)

HpA (F) = HpA (B), HSp (F) = HppA (B), with equivalent norms, respectively.

For p e (0, ot), let Hsp be the set of all f e M such that \\S(f)\\L < ot. Let \\f\\HS = \\S(/)\\L . Note that if $(r) =

r-1/p and Q e B0, then Hsp4 = Hp and \\f\\Hp4, = \\/\\h^. The following is well known as Burkholder's inequality.

Theorem 10 (Burkholder [20]). Ifp e (l, ot), then there exist positive constants Cp and Cp, that depend only on p, such that

cm < ii/iu < cM

for all f 6 10 c M.

For expressions of the constants Cp and Cp, see, for example, [21-23]. See also [24] for Burkholder's inequality on Banach functions spaces.

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

Theorem 11. Let B c F, p e (l, ot) and $ : (0, l] ^ (0, ot). Then

'p\\f\\*„ < \\f\H, < Cp\\f\L , (21)

t+v^K* < \rnK <

< (2Cp + 1

l- (22)

for all f e L1 c M, where Cp and Cp are the constants in Theorem 10.

Next we give the relations between LpA and Lp ' and between Hp ' and Hp-'. We consider the following condition on B:

e Q : En-1 [xB] (to) > 0} e Bn_1 VB e Bn (n>l).

Theorem 12. Let B c F and $ : (0, l] ^ (0, ot). Then

LpA 3 LpA with

Hp4 3 HpP4 with

<2\\f\\L- farpe[1,œ),

<\\f\\H-, forpe(0,œ).

Conversely, if F is regular, B satisfies (23), and $ satisfies the doubling condition, then there exists a positive constant C, dependent only on p, the regularity constant of F, and the doubling constant of <p, such that

Lp' c vm c\\/\\lp, > WfW^ for pe[l,от),

HpA c HSp with C\\f\\HU > \\f\\H% for p e (0, ot) .

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

Theorem 13. Suppose that every a-algebra Fn is generated by countable atoms. Let B = A, p e [l, ot), and $ : (0, l] ^ (0, ot). Assume that $ satisfies the doubling condition and there exists a positive constant C' such that

f1 ^dt<C'A(r) (0<r<1). (28)

Then, there exists a positive constant C such that

2nJ 11lp-* -

L <C\\f\\L- VfeL0v L4 ^ 1 (29)

<c\\fLs- vf e m.

Moreover, if F is regular, then \\f\\L , \\f\\L ,and\\f\\L- are equivalent to each other, and \\/\\Hs , \\/\\Hs and \\/\\Hs- are equivalent to each other.

Using Theorems 11-13, we have Burkholder-type equivalence for generalized martingale Morrey spaces.

Corollary 14. Suppose that every a-algebra Fn is generated by countable atoms. Let B = A, p e (l, ot), and $ : (0, l] ^ (0, ot). Assume that $ satisfies the doubling condition and there exists a positive constant C' such that

f ^dt<C'^(r) (0<r<1).

If F is regular, then there exist positive constants c and C such that, for all f e L^,

* <c\\f\L,■

For the martingale BMO spaces based on square functions, the John-Nirenberg-type equivalence was established by Weisz [25] and [19, Theorem 2.50]. We extend this theorem to the spaces Hp4 and Hp-'.

Theorem 15. Let A(Fn) U A(Fnt c Bn c Fn (n > 0), p e (0, ot) and 0 : (0, l] ^ (0, ot). Assume that $ is almost increasing and satisfies the doubling condition. Then, \\/\\Hs <

<^p,q4\\f\\^- for all q e (0, ot). If we further assume that F is

regular, then \\/\\Hs and \\/\\Hs- are equivalent to \\/\\Hs .

4. Fractional Integrals

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 7.

Let (yn)n>0 be a sequence of nonnegative bounded functions adapted to F = {Fn}n>0; that is, yn is Fn-measurable for every n > 0. Let Iy be the martingale transform associate to (Yn)n>0~; that is,

(Iyf)n =

with convention y-1d0f = 0. Note that if f = (fn)n>0 e M, then Iyf=((Iyf)n)n>o e M.

We now define a generalized fractional integral Ip for martingales as a special case of Iy under the assumption that every a-algebra Fn is generated by countable atoms. Our definition generalizes the fractional integral for dyadic martingales introduced in [6, 7]. The idea of Ip comes from [26].

Suppose that every a-algebra Fn is generated by countable atoms. Let bn be an Fn-measurable function such that

bn (w) = P (B) for a.s. w eB with Be A (Fn); (33) that is,

bn = T P(B)Xb a.s.

BeA(f„)

For a bounded function p : (0,1] ^ (0, rn), we define a generalized fractional integral Ipf = ((Ipf)Jn>0 of f = (fn)n>0 e M by

(Ipf)n = Ip(h-i)dkf-

The generalized fractional integral Ip is obtained by taking yn = p(bn) in (32). If p(r) = ra, a > 0, then we simply denote

Ip by 4.

For quasinormed spaces M1 and M2 of martingales, we denote by B(M1tM2) the set of all bounded martingale transforms from M1 to M2; that is, T e B(M1, M2) means that there exists a positive constant C such that

\\Tf\\M2 <C\\f\\Ml (36)

for all martingales f = (fn)n>0 e M1.

We first study the boundedness on the spaces Hsp^. On martingale Campanato-Hardy spaces, we consider the fractional integral as a martingale transform associated with monotone multipliers. We say a sequence of nonnegative measurable functions y = (yn)^0 is almost decreasing if there exists a positive constant C such that

yk (w) < Cye (w) a.s. Vk > 1. (37)

For an almost decreasing sequence y = (yn)n>0, we define Ay by

A = inf {C > 0 : C satisfies (37)}.

In Theorem 16 below, we do not need any assumption on

{Fn}n>0 •

Theorem 16. Let B c F, p e (0,rn), and : (0,1] ^ (0, rn). Let (yn)n>0 be a sequence of nonnegative bounded almost decreasing adapted functions, and let Iy be the martingale transform defined by (32). Assume that

Cy4v = sup sup

$(P(B))

n>0BeB„ Y(P(B))

\\YnXB\\

Iy £B{

P4'Hpv)

yJ \\k„,

^ AyCy4,V\\

If every a-algebra Fn is generated by countable atoms, then we can apply Theorem 16 to the generalized fractional integral Ip. The following corollary extends [5, Theorem 5.8]

to the spaces Hp ^

Corollary 17. Assume that every a-algebra Fn is generated

by countable atoms and B = A. Let p e (0, x) and

(0,1] ^ (0, x). Suppose that p is almost increasing and that

p(t)<Ht) „

sup- < X.

0<t<l f (t)

Ip eB(Hsp4, HSM).

If one further assumes that {Fn}ni0 is regular and that y is almost increasing and satisfies the doubling condition, then

Ip £B{

U Kv)' (P'<ie(0,x)). (44)

We next study the boundedness on martingale Morrey-Hardy spaces Hp ^ and martingale Hardy spaces Hp.

Recall that A(Fn ^ = Fn for all n>0 if F0 is nonatomic.

Proposition 18. Let B c F, 0 < p < q < x, and $ : (0,1] ^ (0,x). Let (yn)n>0 be a sequence of adapted functions. Suppose that F0 is nonatomic and that B = F. Assume in addition that $ is almost decreasing, that t1/pfy(t) is almost increasing, and that limt^0<p(t) = x. Then, Iy i

vKt'K^ )\{0}.

According to Proposition 18, to consider the boundedness on Hptp and Hs, we suppose that every a-algebra Fn is generated by countable atoms and that B = A.

In this case, if cp(r) = r-1/p and F0 = {Q, 0}, then

. However, if

Hp(^ coincides with Hp and

F0 = {Q, 0}, then Hp ^ does not coincide with Hp in general. We do not always assume that F0 = {Q, 0}.

Theorem 19. Suppose that every a-algebra Fn is generated by countable atoms, that B = A, and that {Fn}n>0 is regular. Let

0 < p < q < œ and 0 : (0,1] ^ (0, œ), and let (yn)n>0 be a sequence of nonnegative bounded adapted functions. Assume that 0 satisfies the doubling condition and that there exists a positive constant C such that

Tyk-i^ (h-i) X{bk tbk_x} (bn)

XTykplX{bt = 1} <c$(bn)pq as

for all n > 0, where bk is the measurable function defined by (33). Then

Iy 6 B {Hp,4' hi<i,pi<i )■ Furthermore, if^(t) = t~1/p, then

Iy 6B(HP

$(r) f k

p(tldt+fi mp(idt<c^

o t It

(0 <r<1).

Ip 6 B iHp,<t>' ) .

The following extends the results for dyadic martingales in [6, 7] and the result for 0 < p < l in [28].

Corollary 21. Suppose that every a-algebra Fn is generated by countable atoms, that B = A, and that {Fn}n>0 is regular. Let 0 < p < q < ot and -l/p + a = -l/q. Then

Ia 6B(HPH).

5. Lemmas

is, <(r) < C0<(s) for all 0 < r < s < 1. Then, forall nonnegative functions F,

As a consequence of Theorem 19, we have the following corollary, which gives an extension of [5, Corollary 5.7] to the spaces Hp' and gives a martingale Morrey-Hardy version of Gunawan [27, Theorem B]:

Corollary 20. Suppose that every a-algebra Fn is generated by countable atoms, that B = A, and that {Fn}n>0 is regular. Let 0 < p < q < ot and : (0,l] ^ (0, ot). Assume that p is bounded, that both p and $ satisfy the doubling condition, and that there exists a positive constant C such that

We prepare some lemmas to prove the results in Sections 3 and 4.

Lemma 1. Let Bn satisfy A(Fn) U A(Fn)± c Bn c Fn(n > 0). Suppose that 0 : (0, l] ^ (0, ot) is almost increasing; that

sup —-—1—— ( —1—- f FdP

bJJ(p(b))\p(b) Jb

< C0 sup

bzbj(p(b))\p(b) Jb

Proof. Let

N = sup / \ „ ( —^ [ FdP) . (52) bJj(p(b))\p(b))b j ( )

For any B e Fn, we can choose the sets Bj, j = 0,l,2,... (finite or infinite) such that

B = UjBj, B0 eA(Fn)±, Bj eA(Fn),

j = 1,2,...

p(b) = tp(bj).

In this case, B: 6 Bn, j = 0,1,2,..., since A(Fn)u A(Fn)± c

Bn. Then

$(P(B))\P(B) Jb 11

4(p(b))p(b)^\)b,

..? U 1

} -(B) U(P(B))p(Bj)iB, , (54)

<TP(Bj)( C0

(B) U(p(Bj))p(Bj)L

= C0N.

This shows the conclusion.

Lemma 2 (see [5, Lemma 3.3]). Suppose that every a-algebra Fn is generated by countable atoms and that {Fn }n>0 is regular. Then, every sequence

B0 DB1 D---DBn Bn eA(Fn), (55)

has the following property: for each n> l,

Bn = B„pi or

(1 + j)p(Bn)<P(Bnpi)<RP(Bn),

where R is the constant in (14).

Lemma 3. Suppose that every a-algebra Fn is generated by countable atoms and that {Fn}n>0 is regular. For B e A(Fm),

let Bj e A(Fj) be

B = Bm c Bm_1 c ■■■ c B0.

Let 0 : (0,1] ^ (0, rn). Suppose that $ satisfies the doubling condition. Then, there exists a positive constant C, thatdepends only on 0 and the regularity constant R, such that

m (be d (t)

I^(bj)x[bj*bhl} <C\ ^dt onB, (58)

j=l+1 Jbm 1

where bj is the function defined by (33).

Proof. Let J = {j : bj = bj_1}. Then, by Lemma 2, we have

I ${bj)x[bj = bM}

j£j,t<j<m

_1_Jhj-! ${bj)

jijl<j<m log (bj-1/bj) Jh *

dt (59)

(hj-1 x J

j£j,e<j<m Jhj

Jh„ t

In Theorem 13, we do not assume that {Fn}n>0 is regular. Hence, we need the following lemma.

Lemma 4. Let $ : (0,1] ^ (0,>x>). Suppose that every aalgebra Fn isgeneratedbycountable atoms. ForB e A(Fn),let Bj e A(Fj) be

B = Bn c Bn-1 c ■■■ c B0.

For the sequence {Bk}k=0 above, one defines a decreasing sequence of integers n.j = nj ({Bk\n=0) inductively by

n1 = sup [ke [0,n]n Z .P (Bk) > 2P (£)}, Hj = sup [k e [0, Hj_1 ] n Z :P (Bk) > 2P (b„h )} (61)

(j>2),

where one uses the convention sup 0 = -1. One further defines

J = {j: nj > 0}, n+ = 1 + nj. (62)

Suppose that $ satisfies the doubling condition. Then, there exists a positive constant C, that depends only on <p, such that

X$(bn+ )<CJ —dt onB

j£j j t

where bj is the function defined by (33).

Note that this lemma is the counterpart to the technique in [29, page 1104, line 5].

Proof. By the definition of nj, if j e J, then

bn- i < bn+ < 2bn._i < bn. on B, (64)

where we use the convention n0 = n.

Using the doubling condition on <p, we have

because the intervals (bn-i, 2bn._i ) are disjointed by (64). □

X*(b, hxf" ^><t (65)

id j i^T->h„:, t Jh„ t

In the proof of Theorem 19, we need the following estimates for the square function of Iyf.

Lemma 5. Suppose that every a-algebra Fn is generated by countable atoms and that {Fn}n>0 is regular. Let p, q e (0, >x) with p < q. Let (yn)n>0 be a sequence of nonnegative bounded adapted functions. Suppose that $ : (0,1] ^ (0,rn) satisfies the doubling condition. Assume that there exists a positive constant C such that

XYk-1$ (bk-1) X\bktbk_x} (bn)

x X Yk-1X[hktk-!} <c$(bn)p/q a.s.

for all n > 0, where bk is the measurable function defined by (33). Then, for f e M with ||/||H = 1,

Sn (IYf)<C(t>(bn_1f\

S(n (lyf)<C$(bn)s/^-1S(f) for all n> 0, where C is a positive constant independent of f. Proof. Let f e M such that HfHHs = 1. We first show that

\dkf\<C4(bk_1), (68)

where C is a positive constant that depends only on 0 and the regularity constant R. Let B e A(Fk). Then, on the set B, keeping in mind that

\dkf\=iml (69)

<^(P(B))\\f\\HS <$(bk-1).

We have obtained (68). We now show (67). Using (68) and the assumption (66), we have

Sn(lyf)2 =Ь1Ш\2

blMh-i)2xlh фь^}

<( Yvk-^ih-i'ixikФък_,} \k=0

< ф(Ьп)2>" < ф(Ъп_1)2р/',

S^f) = IrlMkA2

<S(n) (f) Y yhx^ = hi} k=n+l

<S (f) ( Tyk-iXbФЪк_1)

< $(bnfp/«p2s(n\f)2.

Remark 22. In the course of the proof, the embedding l 1 is used. If one does not use the embedding, then

In-1 \dkf\ < <P(bnp1)p/\

k=0 œ

lYk-i Ш\<Ф(Ьп)РЫ-1^п) (f).

6. Proofs of the Results in Section 3

In this section, we prove the results in Section 3.

Proposition 8 can be proved in the same way as [5, Proposition 2.2], so we omit the proof. Proposition 9 is a direct consequence of Lemma 1. Then, we will prove Theorems 11, 12, and 13.

Recall that S(n)(f) is defined by (11).

6.1. Proof of Theorem 11. We first show Theorem 11, Burkholder's inequality on generalized martingale Campanato spaces.

Proof of Theorem 11. Let f e L01 and Б e Bn. Then, fxB -En[fxB] eL° с M and

dk (fXB - En [fXB]) = {

if к < n, ikf) xb if к > n

Therefore, we have S(fxB - E^fXß]) = S(n\f)XB. Hence, using Theorem 10, we have

Î\i/p

b \f-Enf\pdP) =cp\\fXB -En [fxB]\\Lp <||S(fXB -En [fXB])\L

j S(n\f)pdP

<Cp\\fXB -En [fxB

j \f-Enf\

We have obtained (21).

We next show (22). Using (74), we have

(jB If- En-if\PdpfP < (jB \f- Enf\pdp)

jB \dnf\pdp)

'(j S(n\f)pdpfP

i (75)

(jB\dnf\pdPT

1(jBS(n-1)if)pdp]llp (j S(n-1\f)pdp)llP.

Therefore,

1+Cp " ||Л|2ГМ

For the converse part, using the inequality

(_[ \f- Enf\pdpfp < 2^ \f - EnpJ\pdp)llp, (77)

which we have mentioned in Remark 4, we obtain

^1/p i ( , v \1/p

(j S(n-1\f)pdp) < (j S{n\f)pdp

' j \dnf\pdp

Cp(jB \f-Enf\pdp

j \dnf\pdp

(2Cp + l)(jB \f - En-if\pdp

That is,

\\l-„

< (2Cp + 1

(79) □

6.2. Proof of Theorem 12. We next show Theorem 12, a relation between Lp4 and Lp(f>, H^, and H^.

Proof of Theorem 12. Inequality (24) was mentioned in Remark 4. Inequality (25) is deduced from the inequality

S(f)2 -Sn(f)2 <s(f)2 -Sn-1 (f)2.

We now show (26). Let Be Bn and B' = {u e Q : En-1 [xb](w) > 0}. Since B' e Fn-1, we have

E [Xn\B'XB] = E [Xn\B'En-1 [Xfi]] = 0 (80)

that is, BcB'.

Suppose that {Fn}n>0 is regular. To show (26), we first prove

B' = {ueQ:En-1 [Xb](")>1}, (81)

where R is the regularity constant. By the definition of B', we have B' d {w e Q : En-1[xB](u) > 1/R}. We will show the converse. By the regularity, we have xB < REn-1[xB]. This implies B c e Q : En-1[xB](w) > 1/R}, or equivalently,

xb < x\En_lix«]>1/B}. (82)

Operating En-1, we have

En-1 [xb] < x\E„_1[XB]>1/R}. (83)

We have obtained (81).

From (81), we deduce that

P(B') = E [x\En lto]>1/*}] <E[REn-1 [Xb]] = RP(B).

Hence, using the assumption (23) and the doubling condition on 0 with (84), we have

ml \f-En-J\pdP

^ \f-En-1f\pdP

hi \f-En-1f\pdP

P(B' <t(p(B'))p»^

<t(p(mf\\pLP4.

We have obtained (26).

By the same way as above, we have (27). The proof is completed. □

6.3. Proof of Theorem 13. We now prove Theorem 13, a relation between martingale Morrey spaces and martingale Cam-panato spaces.

Proof of Theorem 13. The part WfWL ^ < 2||/||L was shown in Remark4, and the part ||f||Hs_ < HfH^s is obvious. We now show the part HfHL :

Let B e A(Fn). We take Bk e A(Fk) such that B = Bn c Bn-1 c ••• c B0. Let nj be the decreasing sequence of integers defined in Lemma 4, with convention n0 = n. Since the function En f - En f is constant on Bn. Therefore, on the set B, we have

\En__J-Enj\

wHs-h < fW-L- .

J Kj-^ffdp)

211p( T~ \ J Kf-E„jf\pdp) \p(Bni+ ) V \ J

21/p\\f\\l-j{?{Zn+ )),

where n+ is the same as in (62).

Let J be the same as in Lemma 4 and let m = max J. Using Lemma 4 and the assumption (28), we have

\Enf-Enmf\<X\EnjJ-Enjf\

<WfK X*(p(Bn+ ))

J1 $(*)

P4 ]p(B) t

<\\/\\l- t(P(B)) onB.

For \En f\, we may assume that nm > 0. By the definition of nm,we"have P(B1) < P(B0) < 2P(Bnm) < 2P(B1) < 2P(B0). Therefore,

fP(B0)

<Kp(bÔ)<j .

JP(B0)/2 t

J1 $(t)

< J ^-dt<$(P(B)).

P(B) t

Hence, on the set B, the constant E^f has the following bound:

\ Enm \ = \Enmf Eof\

(ïôb L \Enm f-E0f|'JP)

<2Up\\f\\L-: <p(p(Bi)) < w/wl-^ wm.

Combining (87) and (89), we have

\Enf\ï\\f\\L-4(P(B)) on B.

Using (24) in Theorem 12, we have

H\f\Pdp) <L\f-Enf\Pdp)

+ P(B)i/p \EJ\ <P(B)i/p$(P(B))\\f\\lm + P(B)i/p$(P(B))\\f\\l-

P(B)i/p$(P(B))\

that is,

We can show I

«- by the same way. Indeed, in

(86), we can replace \En, J-En J\ and \\f\\s- by {Sn, (f)2 -

J 1 J J 1

Sn.(f)2]l/2 and \\f\\hs-, respectively. The rest is similar and we can obtain \\/\\hs < \\/\\hs- .

P,4> P4

If B = A, then B satisfies (23). Therefore, if F is regular, we can apply Theorem 12 to obtain the equivalence of \\f\\L ^,

\\f\\l ,, and \\f\\l- and the equivalence of \\/\\hs , \\f\\hs ,

6.4. Proof of Theorem 15. We will now prove Theorem 15, the John-Nirenberg-type theorem for martingale Campanato-Hardy spaces. Following Weisz [19, Definition 2.45], we define

IIbmo? = sup

[S(ff - Snpi(f) for f 6 M and p 6 (0, œ).

Proof of Theorem 15. We may assume that B = F by Proposition 9.

By Holder's inequality and Theorem 12, we only need to show that W/Whs < C A/W^s- for 0 < q < 1 < p.

p, $ r 1 q, 0

Recall the notation S(n\f)2 = S(f)2 - Sn(f)2. Let f e

Hsq4 nL0i c M, A 6 Fn, and m > n + 1. By (73), we have

S(mpl\fXA - EnifxJ) = S(mpl\f)XA. Hence, for B e Fm, m > n + 1,we have

^j/^VXA -En ifXA])qdP

= j S(m-1 (f)SdP P(B)iAnB

< , 1 x f S(mpi)(f)qdP P(AnB)iAnB yJJ

<<KP(AnB))i\\f\\hs-

Therefore, for the {Fm]m>n-martingale (EmifXA] -

EnifXA])m>n,wehaVe

\{Em [fXA] En [/XA])m>n||BMO

<ï(p(a))mhs- .

By [19, Theorem 2.50], there exists a positive constant Cp q4 that depends only on p, q, and 0 such that

{En+i [S(n)(fXA -En [fXA])P])i/P <CM4t(P(A))\\f\U.

Combining (96) and the fact that S(n)(fXA - En[fXA]) = S{n)(f)XA,wehave

(En+i [S(n)(f)p])1/pXA <CM4t(P(A))\\f\\K Therefore, for A e Fn, we have

f 1 c n p \i/p

that is,

\PiA)Ls(n)(f)Pdp

^L^ [Sn)(f)P]dp

<Cp^(P(A))\\f\\hS- ;

< Cp,qMH-

for f e HSp n L°v For general f e HSp, applying (99) to the

martingale f(m) = (/min(m,n))n>0,wehave

II/1U <cp«ar'\u <cp«mht,■ (100)

Taking p = 2 in (100), we have that f is an L2-bounded martingale. Therefore, we have (99) for all f e HThe proof is completed. □

7. Proofs of the Results in Section 4

In this section, we prove the results in Section 4.

7.1. Proofs of Theorem 16 and Corollary 17. Recall that

S(n\f)2 = s(f)2 -Sn(f)2.

Proof of Theorem 16. Using the assumption that (yn)„>0 is almost decreasing, we have

i m S(n\1yf) = I \7k-1dkf\2

< Ayn X \dkf\2

= AynS(n\f)2.

Then, for B e Bn, using the assumption (39), we have \BS(n\lyf)pdP<Apr \Byps(n\f)pdP

<ApYbnXB\L \BS(n\f)pdP <ApY\\YnXB\\pLP(B)ï(P(B))p\\fts,

<ApyCpY^P(B)f(P(B))

p\\f\\p

Therefore, we have

iiviih* <AyCy4Af\]\hsc,

and Iy e B(Hsp4, Hspf). The proofis completed.

Proof of Corollary 17. Let yn = p(bn). Then, we have that (Yn)n>0 is almost decreasing and that I^bUl^ = P(p(B)) for B e A(Fn). Hence,

P(P(B))$(P(B)) C^ = sup sup -W(P(R))-

n>0 BeA(F„) V(F(B))

p(t)<Ht) „

< sup- < œ.

0<t<1 f(t)

Therefore, we can apply Theorem 16 to obtain Iy e B(Hsp4, H ). If we further assume that {Fnjni0 is regular and that y is almost increasing and satisfies the doubling condition, then, by the John-Nirenberg-type equivalence (Theorem 15), we have Iy e B(Hp,4> Hsq f) for all q e (0,rn). ' ' □

Remark 23. Assume that (39) holds. Suppose further that there exists a positive number C' such that 'Z'f=n+1 Yk-1 < C'Yn a.s. for all n > 0. Then, in the light of Remark 22, we see that

s 1/ s

^^^„^(PW)

Y&VÏÏJ \\Hsp4

7.2. Proofs of Theorem 19 and Corollaries 20 and 21

Proof of Theorem 19. We first show the part Iy e B(HSp4, Hsqr,q). Assume (45). Let f = (fj^ e M such that HfHHs = 1. We need only to show that there exists C > 0 independent of f such that

yJ IIhs

To obtain (106), we first show that

s(iyî)<cs(j)

Let N = Xty'h )<S(f)}. We define measurable subsets Q1, Q.2, and Q.3 by

Q1 = {N = œ}, Q2 = {N = 0}, n3 = {0 <N <œ}.

Let w e Q1. Then, we can take infinitely many integers n such that <p{bn_1(w)) < S(f)(w). For such n, we have

Sn (Iyf) (w) < C<p(bn_i (w))piq < CS (f) (w)p/i (109)

by Lemma 5. Letting n ^ >x> along n that satisfies $(bn-1 (w)) < S(f)(w), we have (107) on Q1. On Q2, again by Lemma 5, we have

S(iyf) < C4(b0)pi^-1s(f) < Cs(f)p"-1s(f) = cs(f)pPq.

Let w e Q3. Then, we can take an integer n such that $(bn_1 (w))<S(f)(w), $(bn (w))>S(f)(w). (111)

Hence, by Lemma 5, we have

S(lyf) (w)<Sn (lvf)(w)

+ S(n) {lvf)(<o)

<$(bn-1 (a,))p/«

+ 4(bn (w))p/qp1s(f)(w) (112)

<S(f)(w)p/q + S(f)(cv)p/qp1S(f)(cv)

<S(f)(w)p/q.

We have obtained (107).

We now show (106). Let B e UnA(Fn). Using (107), we have

I, s{l'f)

f S(f)'dP

<p(B)i/q$(p(B))p/q\\f\\pHq

= P(B)i/q$(P (B))p/q.

= +pi/P

We have obtained (106).

We now show the part Iy e B(Hp, HS). Let 0(t) = t~ We simply denote Hp4 by Hp^. Let f = (fn)n>0 e M

suchthat \\f\\ps = 1. Observe that \\f\\ps < ||/||hS .By

p-1/p P

= fpi/p

the assumption that (45) holds for <p(t) = t /p, we can apply (107) to f/\\f\\Hs , and we have

p,-iip

s{iyf)<cs(f)

p/q\\ f^-pp/q

< CS(f)p/q. (114)

Hence, we obtain

The proofis completed

f S(f)PdP Jn

= 1. (115)

Proof of Corollary 20. Let yn = p(bn). We only have to verify (45). Using Lemma 3 and the assumption (48), we have

Tp(bkpi)^{bkpi)X{bt = bk-i}

$(bn) T P(bkpi)X{bk = h-i}

i"1 P(t)$(t)

+HK) jb ^dt<mplS-0 t

By Theorem 19, we have the conclusion. □

Proof of Corollary 21. If p(r) = ra and $(t) = t'1lp, then

I"1 P(t)$(t)

dt ~ r°

= rpi/q = cp(r)p/q.

Observing (116) and applying Theorem 19 to Ia, we have the conclusion. □

Remark 24. In the light of Remark 22, we see that 1

sup sup

JUp iJUp ,

n>0 BeA(F„)$(P (B))p/q

P(B)^\k-i < C\\f\\ps ,

I[I\ykpiäkf\) dp)

q i/ q

which is similar to (105).

In words of harmonic analysis, this corresponds to the

embedding Fp^/^^2 ^ Fp^ for 0 < pi < P2 < œ, 0 < q1,q2 < œ, and se R; see [30, Section 2.3] and [30, page 129] for the definition of the space and the above embedding, respectively. It may be interesting to observe that this embedding is translated into the fact that Ia makes functions have bounded variation.

7.3. Proof of Proposition 18. In this subsection, we prove Proposition 18.

Proof of Proposition 18. To prove Iy i B(Hp4, H^ ) \ {0], we only need to show the following for any f = (fn)n>0 e M:

if I 6 B {^Pp^ rf^M ), then X{\yk-i \>0}dkf = 0

(k=1,2,...).

We now show (119) for f = (fn)n>0 6 M. We may assume that P(\ykpi\ > 0)> 0. For ykpi, define

F ({\ykpi\>0})

= {B6 Fk \ Fkpi :P({\ykpi\ >0}nB)>0}.

If F+k({\ykpi\ > 0]) = 0, then the function X{\Yk-1\>o}dkf is Fk-1 -measurable. Therefore, we have

X{\Yk-i\>0}dkf = Ekpi [X{\yk-i\>0}dkf]

= X{\Yk-i\>0}Ekpi [dkf] = 0

To complete the proof of (119), we only have to show the following:

if F+ ({\Yk-1\>O})=0, then Iy *b(H

p& Hi4

Assume that F+({\yk-1 \ > 0}) = 0. We fix B e Fk \ Fk-1 and S > 0 such that P({\yk-1\ > S} n £) > 0. Let B1 = {\yk-1 \ > 5} and B[ = [\yk-1\ > S} n B. Note that B1 e Fk-1, B[ e F+k({\yk_1\>0}),andB[ cBv

To prove (122), we define two decreasing sequences of measurable sets {Bn}'^1 and {B'n}^==1 that satisfy

Bn e Fk-1, P(Bn) = ^~T,

B'n e F+ ({\7k-1\> 0}), B'n cBn

for every n > 1, inductively as follows.

Suppose that we can choose Bn-1 and B'n-1 that satisfy

Bn-1 e Fk-1, P(Bn-1) = on-2

B'n-1 e F+ ({\yk-1\>0}),

Bn-1 c Bn-1 ■

By the assumption that F0 is nonatomic, Fk-1 is also nonatomic. Hence, there exists Bn e Fk-1 such that Bn c Bn-1, p(Bn) = P(K-1)/2 and P(Bn n B'n_1) > 0. Let B' = Bn n B'n-1. Then, we have Bn and B'n with Bn c Bn-1 and B'n c B'n-1 that satisfy (123).

For the set B'n defined above, let gn = xB' - Ek-1[xB' ]. Since Bn is Fk-1 -measurable, we have

0nXn\B„ = (XB'n - Ek-1 [xb'„])XnVB„

= XB'nXn\B„ - Ek-1 [xB'nXn\B„ ]

By (125) and the assumption that t1/pfy(t) is almost increasing, we have, for any A e U™0 Fn,

$(P(A))\P(A)L ^dP

= $(P(A))P(A)1ip (IahB' \9n\ dP 1

$(P(AnBn))P(AnBn)

JAnB„

$(P(AnBn))

We now show that Iy { B(Hp) ^, ). Suppose that Iy e

B(Hp f, ); that is, there exists a positive number C such that

yj llH

< cllfll

for all fensp4.

Since Bn e Fk \ Fk-1 , we have dkgn = gn = 0, and d:gn = 0 for j = k.

Therefore, we have

S (9n) = 9^ S (ly9n) = yk-19n■ For 9n, we take Dn e U™=0Fn such that

0(P(Dn))(P(ßn)L l9n]PdP

1ip 1 > -

2"^nllHU

By (126) , we may assume that Dn c Bn. As a consequence, we have Dn cB1 = {\Yk-1\ > S}, and limn^mP(DJ = 0by (123). Then, using (127) with (128), we have

<t>{PD>n)yiq (P(Dn)L]9XdP

s<KP(Dn))pi"

mil w-19XdP

< X-lry9n||iJs

1nllH°

0 $(P(Dn))

P(Dn 2 C 1

0 $(P(Dn))

bL \9n\'dP

Therefore, we have

P(Dn)tpiq <f.

However, this contradicts p < q and limt^0$(t) = to. We have (122) and hence have (119). □

Conflict of Interests

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

Acknowledgments

The authors are thankful to Professor Masaaki Fukasawa in Osaka University for his kind hint about Remarks 22, 23, and 24. The first author was supported by Grant-in-Aid for Scientific Research (C) (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. The third author was supported by Grant-in-Aid for Young Scientists (B) (no. 24740085), Japan Society for the Promotion of Science.

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