Hardy–Lorentz spaces for B-valued martingales

J. Math. Anal. Appl. 450 (2017) 1401–1420

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Hardy–Lorentz spaces for B-valued martingales Kaituo Liu a,b , Dejian Zhou a,∗ , Yong Jiao a a b

School of Mathematics and Statistics, Central South University, Changsha 410083, China School of Sciences, Hubei University of Automotive Technology, Shiyan 442002, China

a r t i c l e

i n f o

Article history: Received 27 August 2016 Available online 6 February 2017 Submitted by U. Stadtmueller Keywords: Banach-valued martingale Hardy–Lorentz space Atomic decomposition Dual space

a b s t r a c t In this paper we study the Hardy–Lorentz spaces for Banach space valued martingales. The dual spaces are characterized and several martingale inequalities are established. The proofs mainly depend on the classical tool of atomic decompositions. As usual, these conclusions are closely related to the geometrical properties of the underlying Banach spaces. © 2017 Elsevier Inc. All rights reserved.

1. Introduction The study of Banach space valued (B-valued) martingales began with Pisier’s fundamental paper [22]. Since then, B-valued martingale theory has attracted more attentions in last decades. Burkholder in [2] and [3] discussed martingale transforms and differential subordinations for B-valued martingales. Liu [16,17] introduced the p-variation operator and discussed various B-valued martingale inequalities. Recently, Yu [30,29] investigated the dual spaces of Orlicz–Hardy spaces and weak Orlicz–Hardy spaces for B-valued martingales. We refer to the very recent monograph [23] by Pisier for more information on martingales and Fourier analysis in Banach spaces. It is well known that Lorentz spaces are more extensive family than Lebesgue spaces (see e.g. [21,4]). In [11], the Hardy martingale spaces are extended to Hardy–Lorentz martingale spaces. Very recently, Jiao et al. [13] studied the small-index Hardy–Lorentz martingale spaces and established the predual and John– Nirenberg inequalities for the generalized BMO spaces. Weisz [26] characterized the dual of multi-parameter martingale Hardy–Lorentz spaces. In this paper, our main purpose is to study Hardy–Lorentz spaces for B-valued martingales and extend the dual results in [13] and martingale inequalities in [11] to the B-valued martingale setting. Our proof mainly depends on the establishment of atomic decompositions of Hardy–Lorentz spaces for B-valued mar* Corresponding author. E-mail addresses: [email protected] (K. Liu), [email protected] (D. Zhou), [email protected] (Y. Jiao). http://dx.doi.org/10.1016/j.jmaa.2017.01.093 0022-247X/© 2017 Elsevier Inc. All rights reserved.

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tingales. Recall that atomic decompositions were first introduced by Herz [7], generalized by Weisz [24,25] and developed by many other authors (see e.g. [14,8]) in scalar-valued martingale case. As for B-valued martingales, Liu et al. [19,20] investigated the atomic decompositions and characterized some geometrical properties of Banach spaces; Yu [28] established the dual of B-valued martingale Hardy spaces with the help of atomic decompositions. The results above, and also many other B-valued martingale results (see e.g. [22,9,10,12,23]), are closely related to the geometrical properties of the underlying spaces. Our conclusions in this paper have no exception. The paper is organized as follows. Some preliminary lemmas and B-valued martingale Hardy–Lorentz spaces are introduced in Section 2. In Section 3, we present atomic decompositions for B-valued martinp p gale Hardy–Lorentz spaces Hrs1 ,r2 (B), QSr1 ,r2 (B) and Dr1 ,r2 (B). As usual, these theorems depend on the geometrical properties of the underlying Banach space B. Applying atomic decompositions established in Section 3, we prove two duality results in Section 4. In the last section, we establish several martingale inequalities among Lorentz spaces. These are new versions of the basic inequalities in B-valued martingale setting. Throughout this paper, the sets of integers, nonnegative integers and complex numbers are always denoted by Z, N and C, respectively. We use C to denote a positive constant which may vary from line to line. The symbol ⊂ means the continuous embedding and f ∼ g stands for C −1 g ≤ f ≤ Cg. We call f is equivalent to g if f ∼ g. 2. Preliminaries and notations Let (Ω, F, P) be a complete probability space and B denote a Banach space with norm  · . For a measurable function f , we define its distribution function by   λs (f ) = P {ω ∈ Ω : f (ω) > s} ,

s ≥ 0.

And denote by f ∗ (t) the decreasing rearrangement of f , defined by f ∗ (t) = inf{s ≥ 0 : λs (f ) ≤ t},

t ≥ 0,

with the convention that inf ∅ = ∞. Let 0 < p < ∞, 0 < q ≤ ∞. The B-valued Lorentz space Lp,q (B) (briefly denoted by Lp,q in the sequel) consists of those B-valued measurable functions f with finite quasinorm f p,q given by

f p,q

 q1  ∞ q  p1 ∗ q dt = , t f (t) p t

0 < q < ∞,

0

f p,∞ = sup t p f ∗ (t), 1

q = ∞.

t>0

If 1 < p < ∞ and 1 ≤ q ≤ ∞, the quasinorm  · p,q is equivalent to a norm (see [6, Exercise 1.4.3]). Another equivalent definition of f p,q is given by

f p,q

1  ∞  q dt q 1 = q , tP(f  > t) p t

0 < q < ∞,

0 1

f p,∞ = sup tP(f  > t) p , t>0

q = ∞.

K. Liu et al. / J. Math. Anal. Appl. 450 (2017) 1401–1420

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We now introduce B-valued martingale Hardy–Lorentz spaces. Let {Fn }n≥0 be a nondecreasing sequence of sub-σ-algebras of F such that F = σ( n Fn ). The expectation operator and the conditional expectation operator relative to Fn are denoted by E and En , respectively. A sequence of B-valued random variables f = (fn )n≥0 is called a martingale if En (fn+1 ) = fn for n ≥ 0. Let f = (fn )n≥0 be a B-valued martingale adapted to {Fn }n≥0 such that f0 = 0. For n ≥ 0, denote its martingale difference by dfn = fn − fn−1 (with convention f−1 = 0). For 1 ≤ p < ∞, the maximal function, the p-variation and the conditional p-variation of a B-valued martingale f are respectively defined by Mn (f ) = sup fi ,

M (f ) = sup fn ;

1≤i≤n

Snp (f ) =

n 

dfi p



1 p

,

n≥1

S p (f ) =

i=1

spn (f ) =

n 

∞ 

dfi p

 p1

;

i=1

Ei−1 dfi p

 p1

,

sp (f ) =

i=1

∞ 

Ei−1 dfi p

 p1

.

i=1

Let Γ be the set of all sequences (λn )n≥0 of nondecreasing, nonnegative and adapted functions. Set λ∞ = limn→∞ λn . For 0 < r1 < ∞, 0 < r2 ≤ ∞, 1 ≤ p < ∞, we define B-valued martingale Hardy–Lorentz spaces as follows:  Hr1 ,r2 (B) = f = (fn )n≥0 : f Hr1 ,r2 (B) = M f r1 ,r2 < ∞ ;  p Hrs1 ,r2 (B) = f = (fn )n≥0 : f Hrsp,r (B) = sp (f )r1 ,r2 < ∞ ; 1 2  Sp Hr1 ,r2 (B) = f = (fn )n≥0 : f HrSp,r (B) = S p (f )r1 ,r2 < ∞ ; 1 2  p QSr1 ,r2 (B) = f = (fn )n≥0 : ∃(λn ) ∈ Γ, s.t. Snp (f ) ≤ λn−1 , n ≥ 1, λ∞ ∈ Lr1 ,r2 , where f QSr p,r 

1

2

(B)

= inf λ∞ r1 ,r2 < ∞; (λn )∈Γ

Dr1 ,r2 (B) = f = (fn )n≥0 : ∃(λn ) ∈ Γ, s.t. fn  ≤ λn−1 , n ≥ 1, λ∞ ∈ Lr1 ,r2 , where f Dr1 ,r2 (B) = inf λ∞ r1 ,r2 < ∞. (λn )∈Γ

Taking r1 = r2 = r in the definition above, we obtain the usual B-valued martingale Hardy spaces Hr (B), p p p Hrs (B), HrS (B), QSr (B) and Dr (B) (see [18]). For a Banach space B and t > 0, we recall that the modulus of uniform smoothness is defined as ρB (t) = sup{2−1 (x + ty + x − ty) − 1 : x = y = 1}, and the modulus of uniform convexity is defined as δB (t) = inf{1 − 

x+y  : x = y = 1, x − y = t}. 2

Definition 2.1. We say that a Banach space B is p-uniformly smooth if there is a constant c > 0 such that ρB (t) ≤ ctp for all t > 0. And we say that a Banach space B is q-uniformly convex if there is a constant c > 0 such that δB (t) ≥ ctq for all 0 < t ≤ 2.

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The following two lemmas were proved by Hoffmann-Jørgensen and Pisier [9] (see also [23, Corollary 10.23 and Theorem 10.59]). They reveal the facts that the B-valued martingale inequalities are related closely to the geometrical properties of the underlying spaces. We refer the reader to [15,23] for more information about Radon–Nikodým property (in short R–N property), uniform convexity and uniform smoothness. Lemma 2.2. Let B be a Banach space, 1 < p ≤ 2. Then the following are equivalent: (i) B is isomorphic to a p-uniformly smooth space; (ii) There exists a constant Cp > 0 only depending on p such that En (fm − fn p ) ≤ Cp En

m 

 dfi p ,

∀ 0 ≤ n ≤ m,

i=n+1

for every B-valued martingale f = (fn )n≥0 ;

∞ 

(iii) For every B-valued martingale f = (fn )n≥0 , if E

 dfn p < ∞, then f = (fn )n≥0 converges in

n=1

probability; (iv) There exists a constant C such that for every martingale f = (fn )n≥0 and 1 ≤ r < ∞, M (f )r ≤ CS p (f )r . Lemma 2.3. Let B be a Banach space, 2 ≤ q < ∞. Then the following are equivalent: (i) B is isomorphic to a q-uniformly convex space; (ii) There exists a constant Cq > 0 only depending on q such that En

m 

 dfi q ≤ Cq En (fm − fn q ),

∀0 ≤ n ≤ m,

i=n+1

for every B-valued martingale f = (fn )n≥0 ; (iii) For every uniformly bounded B-valued martingale f = (fn )n≥0 , S q (f ) < ∞, a.e. Remark 2.4. ([23, Corollary 10.7 and 10.23]) Replacing B-valued martingale by dyadic B-valued martingale in (ii) of the two lemmas above, the equivalences still hold (see Section 5 for the definition of B-valued dyadic martingales). We conclude this section with two lemmas which are very useful to verify that a function is in Lorentz spaces Lp,q (Ω) (see [1, Lemma 1.1 and 1.2]). Lemma 2.5. Let 0 < p < ∞, 0 < q ≤ ∞. Assume that the nonnegative sequence {μk } satisfies {2k μk } ∈ lq . Further suppose that the nonnegative function ϕ verifies the following property: there exists 0 < ε < 1 such that, given an arbitrary integer k0 , we have ϕ ≤ ψk0 + ηk0 , where ψk0 is essentially bounded and satisfies ψk0 ∞ ≤ C2k0 , and 2k0 εp P(ηk0 > 2k0 ) ≤ C

∞

k=k0

Then ϕ ∈ Lp,q (Ω) and ϕp,q ≤ C{2k μk }lq .

(2kε μk )p .

K. Liu et al. / J. Math. Anal. Appl. 450 (2017) 1401–1420

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Lemma 2.6. Let 0 < p < ∞, and let the nonnegative sequence {μk } be such that {2k μk } ∈ lq , 0 < q ≤ ∞. Further, suppose that the nonnegative function ϕ satisfies the following property: there exists 0 < ε < 1 such that, given an arbitrary integer k0 , we have ϕ ≤ ψk0 + ηk0 , where ψk0 and ηk0 satisfy k0

2k0 p P(ψk0 > 2k0 )ε ≤ C

(2k μεk )p ,

k=−∞

2k0 εp P(ηk0 > 2k0 ) ≤ C

∞

q 0 < ε < min(1, ), p (2kε μk )p .

k=k0

Then ϕ ∈ Lp,q (Ω) and ϕp,q ≤ C{2k μk }lq . 3. Atomic decompositions In this section, we establish atomic decompositions for B-valued martingale Hardy–Lorentz spaces. Firstly, we recall the definition of an atom. Let T be the set of all stopping times relative to {Fn }n≥0 . For a B-valued martingale f = (fn )n≥0 and a stopping time ν ∈ T , we denote the stopped martingale by f ν = (fnν )n≥0 = (fn∧ν )n≥0 . Definition 3.1. Let 1 ≤ p < ∞, 0 < r < ∞. A B-valued measurable function a is called an atom of the type (1, r, ∞; p) ((2, r, ∞; p) or (3, r, ∞), respectively), if there exists a stopping time ν ∈ T (ν is called the stopping time associated with a) such that (i) an = En a = 0, ∀ n ≤ ν; 1 (ii) sp (a)∞ (S p (a)∞ or M (a)∞ , respectively) ≤ P(ν < ∞)− r . Theorem 3.2. Let B be a Banach space, 1 < p ≤ 2, 0 < r1 ≤ p and 0 < r2 ≤ ∞. Then the following statements are equivalent: (i) B is isomorphic to a p-uniformly smooth space; p (ii) For every f = (fn )n≥0 ∈ Hrs1 ,r2 (B), there exist a sequence (ak )k∈Z of (1, r1 , ∞; p)-atoms and a sequence 1

(μk )k∈Z ∈ lr2 of positive numbers satisfying μk = 3 · 2k P(νk < ∞) r1 (where νk is the stopping time associated with ak ) such that fn =



μk akn ,

a.e.,

∀n ≥ 0,

(3.1)

k∈Z

and f Hrsp,r 1

2

(B)

∼ inf (μk )k∈Z lr2 ,

sup M (ak )r1 < ∞, k∈Z

where akn = En ak and the infimum is taken over all the decompositions of (3.1). p

Proof. (i) ⇒ (ii). Let f = (fn )n≥0 ∈ Hrs1 ,r2 (B). For each k ∈ Z, the stopping time is defined as follows νk = inf{n ∈ N : spn+1 (f ) > 2k },

(inf ∅ = ∞).

Obviously, νk is nondecreasing, νk → ∞ as n → ∞ and sp (f νk ) = spνk (f ) ≤ 2k . Moreover, for n ∈ N,

K. Liu et al. / J. Math. Anal. Appl. 450 (2017) 1401–1420

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(fnνk+1 − fnνk ) =

k∈Z

n



k∈Z

=

n

χ{m≤νk+1 } dfm −

m=0

n

 χ{m≤νk } dfm

m=0

  χ{m≤νk+1 } − χ{m≤νk } dfm = fn .

m=0 k∈Z

For k ∈ Z, let ν

1

μk = 3 · 2k P(νk < ∞) r1 ,

akn =

fnk+1 − fnνk μk

(if μk = 0, set akn = 0).

Then for any fixed k ∈ Z, En−1 dakn =

1 (En−1 dfnνk+1 − En−1 dfnνk ) = 0, μk

∀ n ≥ 0.

So ak = (akn )n≥0 is a B-valued martingale. Since sp (ak ) = 0 on the set {νk = ∞}, by the sublinearity of the operator sp (·), we get 1 sp (f νk+1 ) + sp (f νk ) χ{νk <∞} ≤ P(νk < ∞)− r1 χ{νk <∞} , μk

sp (ak ) = sp (ak χ{νk <∞} ) ≤

which implies sp (ak )∞ ≤ P(νk < ∞)− r1 . By Lemma 2.2(iv), we have 1

M (ak )p ≤ CS p (ak )p = Csp (ak )p ≤ CP(νk < ∞) p − r1 . 1

1

Then for each k ∈ Z, ak is Lp bounded. Consequently, by the R–N property of B (condition (i) implies B has the R–N property; see [18,23]), akn converges a.e. as n → ∞ (denote the limit of akn still by ak ) and akn = En ak for all n ≥ 0. Furthermore, if n ≤ νk , akn = 0. Thus ak is a (1, r1 , ∞; p)-atom and (3.1) holds. Moreover, for a fixed k ∈ Z, we get     r1 r1 E(M (ak )r1 ) = E M (ak )r1 χ{νk <∞} ≤ E(M (ak )p ) p P(νk < ∞)1− p p

≤ CP(νk < ∞)(− r1 +1)

r1 p

r1

P(νk < ∞)1− p = C,

which implies supk∈Z M (ak )r1 < ∞. Now we estimate (μk )lr2 . For 0 < r2 < ∞, since {νk < ∞} = {sp (f ) > 2k }, we get 

|μk |r2

 r1

2

=3



k∈Z

r2

2kr2 P(νk < ∞) r1

 r1

2



=3

k∈Z

≤C

r2

2kr2 P(sp (f ) > 2k ) r1

k∈Z



k 2

r2

y r2 −1 dyP(sp (f ) > 2k ) r1

 r1

2

k∈Z k−1 2

 2

k

≤C

r2

y r2 −1 P(sp (f ) > y) r1 dy

k∈Z k−1 2

≤C

 ∞

r2

y r2 −1 P(sp (f ) > y) r1 dy

0 p

≤ Cs (f )r1 ,r2 = Cf Hrsp,r 1

2

 r1

(B) .

2

 r1

2

 r1

2

K. Liu et al. / J. Math. Anal. Appl. 450 (2017) 1401–1420

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For r2 = ∞, 1

(μk )k∈Z ∞ = sup |μk | = 3 · sup 2k P(νk < ∞) r1 k∈Z

k∈Z

1

= 3 · sup 2k P(sp (f ) > 2k ) r1 k∈Z

≤ Csp (f )r1 ,∞ = Cf Hrsp,∞ (B) . 1

Namely, (μk )k∈Z lr2 ≤ Cf Hrsp,r (B) . 1 2 On the other hand, for an arbitrary integer k0 , let f=



μk ak = g + h,

k∈Z

where g =

k0 −1 k=−∞

μk ak and h =

∞ k=k0

μk ak . By the sublinearity of sp (·), we have

0 −1  k

   sp (g)∞ ≤  μk sp (ak )

k=−∞

≤

k

0 −1

∞

k

0 −1

≤

μk sp (ak )∞

k=−∞

μk P(νk < ∞)− r1 1

k=−∞

≤

k

0 −1

3 · 2k = 3 · 2k0 .

k=−∞

Since sp (ak ) = 0 on the set {νk = ∞}, we have {sp (ak ) > 0} ⊂ {νk < ∞}. It follows from sp (h) ≤

∞ p k k=k0 |μk |s (a ) that {sp (h) > 0} ⊂

∞ 

∞ 

{sp (ak ) > 0} ⊂

k=k0

{νk < ∞}.

k=k0

Consequently, for 0 < ε < 1, we get ∞

2k0 εr1 P(sp (h) > 2k0 ) ≤ 2k0 εr1 P(sp (h) > 0) ≤ 2k0 εr1

P(νk < ∞)

k=k0

= 2k0 εr1

∞

2kεr1 P(νk < ∞)2−kεr1

k=k0

≤

∞

∞ 

2kεr1 P(νk < ∞) =

k=k0

k=k0

By Lemma 2.5, we have sp (f ) ∈ Lr1 ,r2 (Ω) and sp (f )r1 ,r2 ≤ C(μk )k∈Z lr2 . Thus f Hrsp,r 1

2

(B)

∼ inf (μk )k∈Z lr2 ,

where the infimum is taken over all the decompositions of (3.1).

1

2kε P(νk < ∞) r1

r1

.

K. Liu et al. / J. Math. Anal. Appl. 450 (2017) 1401–1420

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(ii) ⇒ (i). Assume that f = (fn )n≥0 is a B-valued martingale satisfying

∞

Edfn p < ∞. We have the

n=0

following estimate: sp (f )1 ≤ sp (f )p = S p (f )p =

∞ 

Edfn p

 p1

< ∞.

n=1 p

p

s Then f ∈ H1s (B) = H1,1 (B). So f has the decomposition as (3.1), where (μk )k∈Z ∈ l1 and k supk∈Z M (a )1 < ∞. The latter property implies that (akn )n≥0 is uniformly integrable for every k ∈ Z ((akn )n≥0 converges to the function ak as n → ∞ in L1 (B), see [23, Theorem 2.9]). Hence, for any ε > 0, there exists Nk ∈ N such that Eakm − akn  < ε for m, n ≥ Nk and every fixed k. For any ε > 0, there exists

k0 ∈ Z such that |k|>k0 μk < ε since (μk )k∈Z ∈ l1 . Set N = max|k|≤k0 {Nk }. Thus, for k0 ∈ Z mentioned above and n, m ≥ N , we get





Efm − fn  = E μk akm − μk akn  ≤ μk Eakm − akn  k∈Z



≤

k∈Z

μk Eakm

−

akn 

|k|>k0

≤C

+ 





k∈Z

μk Eakm − akn 

|k|≤k0



μk + sup E akm − akn 

|k|>k0

|k|≤k0

μk

|k|≤k0

< Cε.

(3.2)

Then (3.2) means that (fn )n≥0 is a Cauchy sequence in L1 (B). Hence, (fn )n≥0 is convergent in probability. By Lemma 2.2, B is isomorphic to a p-uniformly smooth space. The proof is complete. 2

m p Remark 3.3. If r2 = ∞, then the sum k=l μk akn converges to fn in Hrs1 ,r2 (B) as m → ∞, l → −∞. Indeed, by the sublinearity of the operator sp (·), we have m

  fn − μk akn  k=l

p

Hrs1 ,r2 (B)

= sp (fn − fnνm+1 + fnνl )r1 ,r2 ≤ sp (fn − fnνm+1 ) + sp (fnνl )r1 ,r2   ≤ C sp (fn − fnνm+1 )r1 ,r2 + sp (fnνl )r1 ,r2 .

ν

ν

It is obvious that sp (fn −fnm+1 ), sp (fnνl ) ≤ sp (fn ) and sp (fn −fnm+1 ), sp (fnνl ) → 0 a.e. as m → ∞, l → −∞. Then, by the controlled convergence theorem, we have sp (fn − fnνm+1 )r1 ,r2 , sp (fnνl )r1 ,r2 → 0 m  

which means fn − μk akn H sp k=l

r1 ,r2 (B)

as m → ∞, l → −∞,

→ 0 as m → ∞, l → −∞.

Theorem 3.4. Let B be a Banach space, 1 < p ≤ 2, 0 < r1 ≤ p and 0 < r2 ≤ ∞. Then the following statements are equivalent: (i) B is isomorphic to a p-uniformly smooth space; p (ii) For every f = (fn )n≥0 ∈ QSr1 ,r2 (B), there exist a sequence (ak )k∈Z of (2, r1 , ∞; p)-atoms and a sequence 1

(μk )k∈Z ∈ lr2 of positive numbers satisfying μk = 3 · 2k P(νk < ∞) r1 (where νk is the stopping time associated with ak ) such that fn =

k∈Z

μk akn ,

a.e.,

∀n ≥ 0,

(3.3)

K. Liu et al. / J. Math. Anal. Appl. 450 (2017) 1401–1420

f QSr p,r 1

2

(B)

∼ inf (μk )k∈Z lr2 ,

1409

sup M (ak )r1 < ∞, k∈Z

where akn = En ak and the infimum is taken over all the decompositions of (3.3). Proof. (i) ⇒ (ii). The proof is similar to the one of Theorem 3.2, so we give it in sketch. Let f = (fn )n≥0 ∈ p QSr1 ,r2 (B). For k ∈ Z, the stopping time νk is defined by νk = inf{n ∈ N : λn > 2k },

(inf ∅ = ∞),

p

where (λn )n≥0 is the sequence in the definition of QSr1 ,r2 (B). Let akn and μk (k ∈ Z) be the same as in the proof of Theorem 3.2. Then we get (3.3), where (ak )k∈Z is a sequence of (2, r1 , ∞; p)-atoms. Moreover, supk∈Z M (ak )r1 < ∞ and (μk )k∈Z lr2 ≤ Cf QSr p,r (B) still hold. On the other hand, let 1

λn =



2

μk χ{νk ≤n} S p (ak )∞ .

k∈Z p Then (λn )n≥0 is a nondecreasing, nonnegative and adapted sequence with Sn+1 (f ) ≤ λn for every n ≥ 0. Given an integer k0 , let (2) λ∞ = λ(1) ∞ + λ∞ ,

where λ(1) ∞ =

k

0 −1

μk χ{νk <∞} S p (ak )∞ , λ(2) ∞ =

k=−∞

∞

μk χ{νk <∞} S p (ak )∞ .

k=k0 (1)

(2)

Repeating the arguments of Theorem 3.2 (replacing sp (g) and sp (h) by λ∞ and λ∞ , respectively) and p applying Lemma 2.5, we obtain f ∈ QSr1 ,r2 (B) and f QSr p,r (B) ∼ inf (μk )k∈Z lr2 , where the infimum is 1 2 taken over all the decompositions of (3.3). (ii) ⇒ (i). Choose B-valued martingale f = (fn )n≥0 such that S p (f ) ∈ L∞ . Then we have S p (f )pp =

∞ p p p p n=0 Edfn  ≤ S (f )∞ < ∞. Let λn = S (f )∞ for n ≥ 0. Obviously, (λn )n≥0 is a nonnegative, p nondecreasing and adapted sequence with Sn (f ) ≤ λn−1 . Thus we get f QS1,1p (B) ≤ S p (f )∞ < ∞, i.e., p

f ∈ QS1,1 (B). Consequently, fn has the decomposition as (3.4) with supk∈Z M (ak )1 < ∞. The rest of the proof is similar to that of (ii) ⇒ (i) in Theorem 3.2. The proof is complete. 2 Theorem 3.5. Let B be a Banach space, 0 < r1 < ∞ and 0 < r2 ≤ ∞. Then the following statements are equivalent: (i) B has the R–N property; (ii) For every f = (fn )n≥0 ∈ Dr1 ,r2 (B), there exist a sequence (ak )k∈Z of (3, r1 , ∞)-atoms and a sequence 1 (μk )k∈Z ∈ lr2 of positive numbers satisfying μk = 3 · 2k P(νk < ∞) r1 (where νk is the stopping time associated with ak ) such that fn =



μk akn ,

a.e.,

∀n ≥ 0,

k∈Z

f Dr1 ,r2 (B) ∼ inf (μk )k∈Z lr2 ,

sup M (ak )r1 < ∞, k∈Z

where akn = En ak , and the infimum is taken over all the decompositions of (3.4).

(3.4)

K. Liu et al. / J. Math. Anal. Appl. 450 (2017) 1401–1420

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Proof. (i) ⇒ (ii). The proof is similar to the one of Theorem 3.4. Let f = (fn )n≥0 ∈ Dr1 ,r2 (B). For k ∈ Z, the stopping time νk is defined by νk = inf{n ∈ N : λn > 2k },

(inf ∅ = ∞),

where (λn )n≥0 is the sequence in the definition of Dr1 ,r2 (B). Let akn and μk (k ∈ Z) be the same as in the proof of Theorem 3.2. For all n ≥ 0, we have akn  = ≤

1 1 fnνk+1 − fnνk  ≤ (λνk+1 −1 + λνk −1 )χ{νk <∞} μk μk 1 1 k+1 (2 + 2k )χ{νk <∞} ≤ P(νk < ∞)− r1 χ{νk <∞} . μk

Hence M (ak )∞ ≤ P(νk < ∞)− r1 . 1

Because B has the R–N property, there exists a B-valued integrable function (still denoted by ak ) such that akn converges to ak as n → ∞ in L1 (B) (see [23, Theorem 2.9]). Then ak is a (3, r1 , ∞)-atom and (3.4) holds. Moreover, we have supk∈Z M (ak )r1 < ∞ and (μk )k∈Z lr2 ≤ Cf Dr1 ,r2 (B) . On the other hand, let λn =



μk χ{νk ≤n} M (ak )∞ .

k∈Z

Then (λn )n≥0 is a nondecreasing, nonnegative and adapted sequence with fn+1  ≤ λn . Given an integer k0 , set (2) λ∞ = λ(1) ∞ + λ∞ ,

where

λ(1) ∞ =

k

0 −1 k=−∞

μk χ{νk <∞} M (ak )∞ , λ(2) ∞ =

∞

μk χ{νk <∞} M (ak )∞ .

k=k0 (1)

(2)

Replacing sp (g) and sp (h) in the proof of Theorem 3.2 by λ∞ and λ∞ and using Lemma 2.5 again, we get f ∈ Dr1 ,r2 (B) and f Dr1 ,r2 (B) ∼ inf (μk )k∈Z lr2 , where the infimum is taken over all the decompositions of (3.4). (ii) ⇒ (i). Take B-valued martingale f = (fn )n≥0 such that supn≥0 fn ∞ < ∞. Let λn = supn≥0 fn ∞ for all n ≥ 0. Then we get f D1,1 (B) ≤ supn≥0 fn ∞ which implies that f ∈ D1,1 (B). Similar to the proof of (ii) ⇒ (i) in Theorem 3.2, we obtain that (fn )n≥0 converges in probability and (fn )n≥0 is uniformly integrable. Then (fn )n≥0 converges in L1 (B). Hence, the space B has the R–N property (see [5] or [18, p. 31]). The proof of the theorem is complete. 2 Remark 3.6. In Theorem 3.2, 3.4 and 3.5, we also prove that if a martingale f has an atomic decomposition 1 with μk = 3 · 2k P(νk < ∞) r1 , then f is in the corresponding Hardy–Lorentz space.

K. Liu et al. / J. Math. Anal. Appl. 450 (2017) 1401–1420

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4. Dual spaces In this section, we prove two duality results. The following lemma will be used. Lemma 4.1. Let 1 < p ≤ 2, 0 < r1 < p and 0 < r2 < ∞. If B is isomorphic to a p-uniformly smooth space, p p then Hps (B) is dense in Hrs1 ,r2 (B). p

Proof. Let f = (fn )n≥0 ∈ Hrs1 ,r2 (B). According to Theorem 3.2, there exist a sequence of (1, r1 , ∞, p)-atoms 1

(ak )k∈Z and a sequence (μk )k∈Z ∈ lr2 of positive numbers satisfying μk = 3 · 2k P(νk < ∞) r1 (where νk is the stopping time associated with ak ) such that

fn =

μk akn =

k∈Z



(fnνk+1 − fnνk )

For an arbitrary positive integer N , set g N = (gnN )n≥0 , where gnN =  spn (g N )

p

=s

∀n ≥ 0.

a.e.,

k∈Z

(gnN )

p

=s

N

 μk akn

 p

=s

k=−N

N

N

k=−N

μk akn for every n ≥ 0. Then 

(fnνk+1

−

fnνk )

k=−N

= sp (fnνN +1 − fnν−N ) ≤ sp (fnνN +1 ) + sp (fnν−N ) ≤ 2N +1 + 2−N < ∞. So we have g N Hpsp (B) = sp (g N )p ≤ 2N +1 + 2−N < ∞. This means g N ∈ Hps (B). By Remark 3.3, p p gnN → fn in Hrs1 ,r2 (B) as n → ∞. Then {g N }N ≥1 converges to f in Hrs1 ,r2 (B). We conclude the proof of the theorem. 2 p

p

To describe the duality of B-valued martingale Hardy–Lorentz spaces Hrs1 ,r2 (B), we introduce the following definition. Definition 4.2. Let 1 ≤ p < ∞ and α ≥ 0. The generalized B-valued martingale BMO space is defined by  p p BM Oαs (B) = f = (fn )n≥0 ∈ Hps (B) : f BM Oαsp (B) < ∞ , where f BM Oαsp (B) = sup P(ν < ∞)− p −α sp (f − f ν )p . 1

ν∈T

Theorem 4.3. Let 0 < r1 , r2 ≤ 1 and 1 < p ≤ 2. If B is isomorphic to a p-uniformly smooth space, then 

∗

p

Hrs1 ,r2 (B)

= BM Oαs (B∗ ), q

q=

Proof. Let g ∈ BM Oαs (B∗ ) ⊂ Hqs (B∗ ). Since f Hrsp,r q

1 p ,α = − 1. p−1 r1

q

= sp (f )r1 ,r2 ≤ sp (f )p,p = f Hpsp (B) , we ∗  p p p q get Hps (B) ⊂ Hrs1 ,r2 (B). Define (refer to [27] for the fact Hps (B) = Hqs (B∗ )) 1

φg (f ) = E(f g),

2

(B)

p

∀f ∈ Hps (B).

Applying Theorem 3.2 and Hölder’s inequality, we have

K. Liu et al. / J. Math. Anal. Appl. 450 (2017) 1401–1420

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|E(f g)| =



 μk E

k∈Z

≤



≤ ≤

daki d(gi − giνk )

∞   p1   q1

μk E Ei−1 daki p Ei−1 d(gi − giνk )q ∞ 

μk E



i=1



Ei−1 daki p

∞  p1 

i=1

k∈Z





i=1



k∈Z



∞

Ei−1 d(gi − giνk )q

 q1



i=1



μk E sp (ak )p χ{νk <∞}

 p1

sq (g − g νk )q

k∈Z

≤



μk P(νk < ∞) p − r1 sq (g − g νk )q 1

1

k∈Z

=



μk P(νk < ∞)− q −α sq (g − g νk )q . 1

k∈Z

It follows from r2 ≤ 1 and Theorem 3.2 that |φg (f )| ≤ C{μk }lr2 gBM Oαsq (B∗ ) ≤ Cf Hrsp,r 1

2

s (B∗ ) . (B) gBM Oα q

p

By Lemma 4.1, φg can be uniquely extended to a continuous functional on Hrs1 ,r2 (B).  p ∗  p ∗ p p q Conversely, let φ ∈ Hrs1 ,r2 (B) . Since Hps (B) ⊂ Hrs1 ,r2 (B) and Hps (B) = Hqs (B∗ ), there exists q g ∈ Hqs (B∗ ) such that φ(f ) = E(f g),

p

∀f ∈ Hps (B).

For any stopping time ν ∈ T , set b = g − g ν . Then b ∈ Hqs (B∗ ). Since B is isomorphic to a p-uniformly  q ∗ p smooth space, then B is reflexive [15, Proposition 1.e.3]. Thus we obtain Hqs (B∗ ) = Hps (B). So for any p ε > 0, there exists some aε ∈ Hps (B) with aε Hpsp (B) < 1 + ε such that q

    bHqsq (B∗ ) = |E (g − g ν )aε | = |E g(aε − aνε ) |.   p In particular, there exists a ∈ Hps (B) with aHpsp (B) ≤ 1 such that bHqsq (B∗ ) = |E g(a − aν ) |. Set h=

a − aν 2P(ν < ∞) r1 − p 1

Since 0 < r1 , r2 ≤ 1, there exist R1 , R2 > 0 such that obtain hHrsp,r 1

2

(B)

= ≤

1 2P(ν < ∞)

1 r1

1 −p

1 r1

=

1

1 p

.

+ R11 ,

1 r2

=

1 p

+ R12 . By Hölder’s inequality, we

sp (a − aν )r1 ,r2

C

p ν 1 1 s (a − a )p,p χ{ν<∞} R1 ,R2 2P(ν < ∞) r1 − p ⎞ R1 ⎛ 2 ∞   p R R 2 2 Cs (a)p ⎝ R2 ∗ R1 −1 ⎠ ≤ t (t) dt χ 1 1 {ν<∞} P(ν < ∞) r1 − p R1 0

K. Liu et al. / J. Math. Anal. Appl. 450 (2017) 1401–1420

≤

C P(ν < ∞) r1 − p 1

1

⎞ R1 ⎛∞ 2  R2 ⎝ t R1 −1 χ0,P(ν<∞) (t)dt⎠ ⎛

≤

=

C P(ν < ∞) r1 − p 1

1

C P(ν < ∞)

1413

⎜ ⎝

0 P(ν<∞) 

t

⎞ R1

2

R2 R1

−1

⎟ dt⎠

0 1

1 r1

1 −p

P(ν < ∞) R1 = C.

Consequently, we have φ ≥ C −1 |φ(h)| = |E(gh)|

  1 1 = C −1 P(ν < ∞) p − r1 |E g(a − aν ) | = C −1 P(ν < ∞) p − r1 bHqsq (B∗ ) 1

1

= C −1 P(ν < ∞)− q −α sq (g − g ν )q . 1

Taking the supremum over all stopping times, we get gBM Oαsq (B∗ ) ≤ Cφ. The proof is complete. 2 Remark 4.4. If r1 = r2 = r, then Theorem 4.3 reduces to the corresponding result in [28]. p

In order to characterize the dual space of Hrs1 ,r2 (B) for 0 < r1 ≤ 1 and 1 < r2 < ∞, we now introduce the following definition which is motivated by [13]. Definition 4.5. Let 1 ≤ p < ∞, 1 ≤ r < ∞ and α ≥ 0. We define the new generalized B valued martingale BMO spaces as  p sp BM Or,α (B) = f = (fn )n≥0 ∈ Hps (B) : f BM Or,α sp (B) < ∞ , where

f BM Or,α sp (B) = sup

k∈Z

1

2k P(νk < ∞)1− p sp (f − f νk )p  

2k P(νk < ∞)1+α

r  r1

,

k∈Z

 where the supremum is taken over all stopping time sequences {νk }k∈Z such that 2k P(νk < ∞)1+α k∈Z ∈ lr . Theorem 4.6. Let 0 < r1 ≤ 1, 1 < r2 < ∞ and 1 < p ≤ 2. If B is isomorphic to a p-uniformly smooth space, then 

p

∗

Hrs1 ,r2 (B)

= BM Ors2 ,α (B∗ ), q

q=

1 p ,α = − 1. p−1 r1

Proof. Let g ∈ BM Ors2 ,α (B∗ ) ⊂ Hqs (B∗ ). As mentioned in Theorem 4.3, we have Hps (B) ⊂ Hrs1 ,r2 (B). Define q

q

φg (f ) = E(f g),

p

p

∀f ∈ Hps (B).

Applying Theorem 3.2 and Hölder’s inequality, we have

p

K. Liu et al. / J. Math. Anal. Appl. 450 (2017) 1401–1420

1414

|E(f g)| =



 μk E

k∈Z

≤



 daki d(gi − giνk )

i=1



∞ 

μk E

Ei−1 daki p

∞  p1 

i=1

k∈Z

≤

∞

Ei−1 d(gi −

giνk )q

 q1



i=1

  p1 μk E sp (ak )p χ{νk <∞} sq (g − g νk )q

k∈Z



≤3

1

2k P(νk < ∞)1− q sq (g − g νk )q .

k∈Z

By the definition of  · BM Orsq ,α (B∗ ) , we obtain 2

 |E(fn gn )| ≤ 3



2k P(νk < ∞)1+α

r2

 r1

2

gBM Orsq ,α (B∗ ) 2

k∈Z

 ≤3



2k P(νk < ∞)

1 r1

r2

 r1

2

gBM Orsq ,α (B∗ ) 2

k∈Z

≤ Cf Hrsp,r 1

2

(B)

· gBM Orsq ,α (B∗ ) . 2

Consequently, we have |φg (f )| ≤ Cf Hrsp,r

(B) · gBM Ors2 ,α (B∗ ) . By 1 2 sp Hr1 ,r2 (B). p p Since Hps (B) ⊂ Hrs1 ,r2 (B) and

tended to a continuous functional on  p ∗ Conversely, let φ ∈ Hrs1 ,r2 (B) . q g ∈ Hqs (B∗ ) such that

q

φ(f ) = E(f g),

Lemma 4.1, φg can be uniquely ex sp ∗ q Hp (B) = Hqs (B∗ ), there exists

p

∀f ∈ Hps (B). 1

Let {νk }k∈Z be an arbitrary stopping time sequence such that {2k P(νk < ∞) r1 }k∈Z ∈ lr2 . Similar to  p Theorem 4.3, there exists some ak ∈ Hps (B) with ak Hpsp (B) ≤ 1 such that g − g νk Hqsq (B∗ ) = E ak (g −  g νk ) for every k ∈ Z. For N ∈ N, set

h=

N

1

2k P(νk < ∞)1− q (ak − aνkk ).

k=−N

k0 −1 k

N 1 1 Let h = G + H, where G = k=−N 2 P(νk < ∞)1− q (ak − aνkk ), H = k=k0 2k P(νk < ∞)1− q (ak − aνkk ) if −N ≤ k0 ≤ N ; G = 0 if k0 ≤ −N ; H = 0 if k0 > N . Let ε = rp1 (< 1). By Chebyshev’s inequality and the sublinearity of sp (·), we have 2k0 r1 P(sp (G) > 2k0 )ε ≤ 2k0 r1 ≤

0 −1  k

1 sp (G)pε p 2k0 εp 1

2k P(νk < ∞)1− q sp (a − aνk )p

pε

k=−N

≤C

k

0 −1 k=−N

On the other hand,



ε

2k P(νk < ∞) r1

r1

≤C

k

0 −1 k=−∞



ε

2k P(νk < ∞) r1

r1

.

K. Liu et al. / J. Math. Anal. Appl. 450 (2017) 1401–1420

{sp (H) > 0} ⊂

N 

1415

{νk < ∞}.

k=k0

Then for each 0 < ε < 1, we obtain 2k0 εr1 P(sp (H) > 2k0 ) ≤ 2k0 εr1 P(sp (H) > 0) ≤ 2k0 εr1

N

P(νk < ∞)

k=k0

≤

N

2kεr1 P(νk < ∞) ≤

k=k0

≤

∞

N  r1

1 2kε P(νk < ∞) r1 k=k0



1

2kε P(νk < ∞) r1

r1

.

k=k0 1

By Lemma 2.6, we get sp (h) ∈ Lr1 ,r2 and sp (h)r1 ,r2 ≤ C{2k P(νk < ∞) r1 }k∈Z lr2 . Consequently, p h ∈ Hrs1 ,r2 (B) and hHrsp,r 1

(B) 2

≤C

   r1 1 r 2 2 2k P(νk < ∞) r1 . k∈Z

Therefore, N

1

2k P(νk < ∞)1− q sp (g − g νk )q =

k=−N

N

  1 2k P(νk < ∞)1− q E ak (g − g νk )

k=−N

=

N

  1 2k P(νk < ∞)1− q E (ak − aνkk )g

k=−N

= E(hg) = φ(h) ≤ hHrsp,r 1

2

(B) φ

   r1 1 r 2 2 φ. 2k P(νk < ∞) r1 ≤C k∈Z

Thus we get N

1

2k P(νk < ∞)1− q sq (g − g νk )q

k=−N

  r2  r12 2k P(νk < ∞)1+α

≤ Cφ.

k∈Z

Taking over all N ∈ N and the supremum over all of such stopping time sequences satisfying {2k P(νk < 1 ∞) r1 }k∈Z ∈ lr2 , we obtain gBM Orsq ,α (B∗ ) ≤ Cφ. The proof is complete. 2 2

5. Martingale inequalities In this section, as another application of the atomic decomposition, we obtain a sufficient condition for a σ-sublinear operator to be bounded from B-valued martingale Hardy–Lorentz spaces to function Lorentz spaces.

∞ An operator T : X → Y is called a σ-sublinear operator if for any α ∈ C it satisfies |T ( k=1 fk )| ≤

∞ k=1 |T (fk )| and |T (αf )| ≤ |α||T (f )|, where X is a martingale space and Y is a measurable function space.

K. Liu et al. / J. Math. Anal. Appl. 450 (2017) 1401–1420

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p

Lemma 5.1. Let 1 < p ≤ 2 and let T : Hps (B) → Lp (Ω) be a bounded σ-sublinear operator. If B is isomorphic to a p-uniformly smooth space and {|T a| > 0} ⊂ {ν < ∞} for all (1, r1 , ∞; p)-atoms a (where ν is the stopping time associated with a), then for 0 < r1 < p and 0 < r2 ≤ ∞, we have T f r1 ,r2 ≤ Cf Hrsp,r 1

2

(B) .

p

Proof. Let f ∈ Hrs1 ,r2 (B). By Theorem 3.2, f can be decomposed into the sum of a sequence of (1, r1 , ∞; p)-atoms. For k0 ∈ Z, set

f=



k

0 −1

k

μk a =

k

∞

k

μk a +

k=−∞

μk ak = g + h,

k=k0

where g=

k

0 −1

k

μk a ,

h=

k=−∞

∞

μk a k .

k=k0

Let ε = r1 /p. By Chebyshev’s inequality and the sublinearity, boundedness of T , we get  1 ε p T (g) p 2k0 p k −1 0 0 −1 

pε  k

pε ≤ μk T (ak )p ≤ μk sp (ak )p

2k0 r1 P(T (g) > 2k0 )ε ≤ 2k0 r1

k=−∞



k=−∞

k

0 −1

≤C

pε

μk P(νk < ∞)

1 1 p − r1

k=−∞

=C

k

0 −1

k

0 −1  k  k ε r ε r 1 1 2 P(νk < ∞) r1 ≤C 2 P(νk < ∞) r1 .

k=−∞

On the other hand, we have {T (h) > 0} ⊂

k=−∞ ∞

{νk < ∞}. Then for each 0 < ε < 1, we obtain

k=k0

2k0 εr1 P(T (h) > 2k0 ) ≤ 2k0 εr1 P(T (h) > 0) ≤ 2k0 εr1

∞

P(νk < ∞)

k=k0

≤

∞

2

kεr1

∞

 kε 1 r 1 P(νk < ∞) = 2 P(νk < ∞) r1

k=k0

≤

∞



k=k0 1

2kε P(νk < ∞) r1

r1

.

k=k0

By Lemma 2.6 and Theorem 3.2, we get T (f ) ∈ Lr1 ,r2 (B) and 1

T (f )r1 ,r2 ≤ C{2k P(νk < ∞) r1 }k∈Z lr2 ≤ Cf Hrsp,r 1

2

(B) .

2

Similar to the proof of Lemma 5.1, the following two lemmas can be proved by using Theorem 3.4 and 3.5, respectively. We omit the proofs.

K. Liu et al. / J. Math. Anal. Appl. 450 (2017) 1401–1420

1417

p

Lemma 5.2. Let 1 < p ≤ 2 and let T : HpS (B) → Lp (Ω) be a bounded σ-sublinear operator. If B is isomorphic to a p-uniformly smooth space and {|T a| > 0} ⊂ {ν < ∞} for all (2, r1 , ∞; p)-atoms a (where ν is the stopping time associated with a), then for 0 < r1 < p and 0 < r2 ≤ ∞, we have T f r1 ,r2 ≤ Cf QSr p,r 1

2

(B) .

Lemma 5.3. Let 0 < q < ∞ and let T : Hq (B) → Lq (Ω) be a bounded σ-sublinear operator. If B has the R–N property and {|T a| > 0} ⊂ {ν < ∞} for all (3, r1 , ∞)-atoms a (where ν is the stopping time associated with a), then for 0 < r1 < q and 0 < r2 ≤ ∞, we have T f r1 ,r2 ≤ Cf Dr1 ,r2 (B) . Let ([0, 1), F, μ) be a probability space such that μ is the Lebesgue measure and subalgebras {Fn }n≥0 generated as follows: Fn = σ-algebra generated by atoms

 j j + 1 , , j = 0, · · · , 2n − 1. 2n 2n

Recall that all martingales with respect to {Fn }n≥0 are called dyadic martingales. In the sequel, Remark 2.4 will be used. Theorem 5.4. Let B be a Banach space, 1 < p ≤ 2, 0 < r1 < p and 0 < r2 ≤ ∞. Then the following statements are equivalent: (i) B is isomorphic to a p-uniformly smooth space; p (ii) There exists a constant C > 0 such that for every f = (fn )n≥0 ∈ Hrs1 ,r2 (B), M f r1 ,r2 ≤ Cf Hrsp,r 1

2

(B) ; p

(iii) There exists a constant C > 0 such that for every f = (fn )n≥0 ∈ QSr1 ,r2 (B), M f r1 ,r2 ≤ Cf QSr p,r 1

2

(B) .

Proof. (i) ⇒ (ii). The maximal operator T f = M f is σ-sublinear. Since B is isomorphic to a p-uniformly smooth space and [23, Theorem 10.60], we have M f p ≤ Csp (f )p = Cf Hpsp (B) . p

This means M : Hps (B) → Lp (Ω) is bounded. For any (1, r1 , ∞; p)-atom a and the corresponding stopping time ν, we have {|T a| > 0} = {|M a| > 0} ⊂ {ν < ∞}. Hence, P(|T a| > 0) ≤ P(ν < ∞). The desired inequality immediately follows from Lemma 5.1. (i) ⇒ (iii). Similar to (i) ⇒ (ii), it can be proved by Lemma 5.2.  ∞  p (ii) ⇒ (i). Let f = (fn )n≥0 be an arbitrary B-valued martingale with Esp (f )p = E < ∞. n=1 dfn  p p sp Since 0 < r1 < p, we have s (f )r1 ,r2 ≤ s (f )p,p < ∞. So the martingale f ∈ Hr1 ,r2 (B). Consider (n)

(n)

g (n) = (gm )m≥0 , where gm = fm+n − fn , (m ≥ 0). It is obvious that sp (g (n) )p = sp (f )p − spn (f )p → 0 as n → ∞ and sp (g (n) ) ≤ sp (f ). Furthermore, by condition (ii) we have fm+n − fn r1 ,r2 ≤ sup fm+n − fn r1 ,r2 ≤ M g (n) r1 ,r2 ≤ Csp (g (n) )r1 ,r2 . m≥0

K. Liu et al. / J. Math. Anal. Appl. 450 (2017) 1401–1420

1418

Applying the controlled convergence theorem, we obtain {fn }n≥1 is a Cauchy sequence in Lr1 ,r2 (B). Hence fn is convergent in probability. By Lemma 2.2, B is isomorphic to a p-uniformly smooth space.  ∞  p < ∞. (iii) ⇒ (i). Let f = (fn )n≥0 be an arbitrary B-valued dyadic martingale such that E n=1 dfn  p sp Similar to (ii) ⇒ (i), we get f ∈ Hr1 ,r2 (B). For n ≥ 0, let λn = sn+1 (f ). Then (λn )n≥0 is a nonnegative, nondecreasing and adapted sequence. Since f is a B-valued dyadic martingale, we have Snp (f ) ≤ Cspn (f ). Thus f QSr p,r 1

p

2

≤ sp (f )r1 ,r2 < ∞.

(B)

(n)

Namely, f ∈ QSr1 ,r2 (B). Consider g (n) = (gm )m≥0 as above. By condition (iii), we get fm+n − fn r1 ,r2 ≤ M g (n) r1 ,r2 ≤ Cg (n) QSr p,r 1

2

(B)

≤ Csp (g (n) )r1 ,r2 .

Using the controlled convergence theorem, we obtain {fn }n≥1 is a Cauchy sequence in Lr1 ,r2 (B). Hence fn is convergent in probability. By Lemma 2.2 and Remark 2.4, B is isomorphic to a p-uniformly smooth space. The proof of the theorem is complete. 2 Lemma 5.5. ([23, Theorem 10.58]) Let 2 ≤ q ≤ r < ∞. Then the following statements are equivalent: (i) B is isomorphic to a q-uniformly convex space; (ii) There exists a constant C such that for every f = (fn )n≥0 ∈ Hr (B), f HrSq (B) ≤ Cf Hr (B) ; (iii) There exists a constant C such that for every f = (fn )n≥0 ∈ Hr (B), f Hrsq (B) ≤ Cf Hr (B) . Theorem 5.6. Let B be a Banach space, 2 ≤ q < ∞, 0 < r1 < q and 0 < r2 ≤ ∞. Then the following statements are equivalent: (i) B is isomorphic to a q-uniformly convex space; (ii) There exists a constant C > 0 such that for every B-valued martingale f = (fn )n≥0 , f HrSq,r 1

2

(B)

≤ Cf Dr1 ,r2 (B) ;

(iii) There exists a constant C > 0 such that for every B-valued martingale f = (fn )n≥0 , f Hrsq,r 1

2

(B)

≤ Cf Dr1 ,r2 (B) .

Proof. (i) ⇒ (ii). It is obvious that {S q (a) > 0} ⊂ {ν < ∞}, where a is a (3, r1 , ∞)-atom and ν is the corresponding stopping time. By Lemma 5.5, we know that the sublinear operator S q (·) is bounded from Hq (B) to Lq (Ω). Condition (i) implies that the space B has the R–N property. Then by Lemma 5.3, we have S q (f )r1 ,r2 ≤ Cf Dr1 ,r2 (B) , Namely, f HrSq,r (B) ≤ Cf Dr1 ,r2 (B) . 1 2 (i) ⇒ (iii). It can be similarly proved as above.

∀f = (fn )n≥0 ∈ Dr1 ,r2 (B).

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(ii) ⇒ (i). Let f = (fn )n≥0 be an arbitrary B-valued martingale such that supn≥0 fn ∞ < ∞. Then f Dr1 ,r2 (B) < ∞. Since f HrSq,r (B) ≤ Cf Dr1 ,r2 (B) , we have S q (f ) < ∞. By Lemma 2.3, B is isomorphic 1 2 to a q-convex space. (iii) ⇒ (i). Consider a B-valued dyadic martingale f = (fn )n≥0 with supn≥0 fn ∞ < ∞. Then S q (f ) ≤ Csq (f ) < ∞. By Lemma 2.3 and Remark 2.4, we get the desired result. The proof of the theorem is complete. 2 Acknowledgment The authors would like to thank the referee for her/his careful reading and useful comments. References [1] W. Abu-Shammala, A. Torchinsky, The Hardy–Lorentz spaces H p,q (Rn ), Studia Math. 182 (3) (2007) 283–294, MR 2360632. [2] D. Burkholder, Martingale transforms and the geometry of Banach spaces, in: Probability in Banach Spaces, III, Medford, Mass., 1980, in: Lecture Notes in Math., vol. 860, Springer, Berlin–New York, 1981, pp. 35–50, MR 647954. [3] D. Burkholder, Differential subordination of harmonic functions and martingales, in: Harmonic Analysis and Partial Differential Equations, El Escorial, 1987, in: Lecture Notes in Math., vol. 1384, Springer, Berlin, 1989, pp. 1–23, MR 1013814. [4] M. Carro, J. Raposo, J. Soria, Recent developments in the theory of Lorentz spaces and weighted inequalities, Mem. Amer. Math. Soc. 187 (877) (2007) xii+128, MR 2308059. [5] S. Chatterji, Martingale convergence and the Radon–Nikodym theorem in Banach spaces, Math. Scand. 22 (1968) 21–41, MR 0246341. [6] L. Grafakos, Classical Fourier Analysis, second ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008, MR 2445437. [7] C. Herz, Bounded mean oscillation and regulated martingales, Trans. Amer. Math. Soc. 193 (1974) 199–215, MR 0353447. [8] K. Ho, Atomic decompositions, dual spaces and interpolations of martingale Hardy–Lorentz–Karamata spaces, Q. J. Math. 65 (3) (2014) 985–1009, MR 3261978. [9] J. Hoffmann-Jørgensen, G. Pisier, The law of large numbers and the central limit theorem in Banach spaces, Ann. Probab. 4 (4) (1976) 587–599, MR 0423451. [10] Y. Jiao, Carleson measures and vector-valued BMO martingales, Probab. Theory Related Fields 145 (3–4) (2009) 421–434, MR 2529435. [11] Y. Jiao, L. Peng, P. Liu, Atomic decompositions of Lorentz martingale spaces and applications, J. Funct. Spaces Appl. 7 (2) (2009) 153–166, MR 2541232. [12] Y. Jiao, L. Wu, M. Popa, Operator-valued martingale transforms in rearrangement invariant spaces and applications, Sci. China Math. 56 (4) (2013) 831–844, MR 3034845. [13] Y. Jiao, L. Wu, A. Yang, R. Yi, The predual and John–Nirenberg inequalities on generalized BMO martingale spaces, Trans. Amer. Math. Soc. 369 (1) (2017) 537–553. [14] Y. Jiao, G. Xie, D. Zhou, Dual spaces and John–Nirenberg inequalities of martingale Hardy–Lorentz–Karamata spaces, Q. J. Math. 66 (2) (2015) 605–623, MR 3356840. [15] J. Lindenstrauss, L. Tzafriri, Function spaces, in: Classical Banach Spaces, II, in: Ergebnisse der Mathematik und ihrer Grenzgebiete (Results in Mathematics and Related Areas), vol. 97, Springer-Verlag, Berlin–New York, 1979, MR 540367. [16] P. Liu, Martingale inequalities and convexity and smoothness of Banach spaces, Acta Math. Sinica 32 (6) (1989) 765–775, MR 1052220. [17] P. Liu, Martingale spaces and geometrical properties of Banach spaces, Sci. China Ser. A 34 (5) (1991) 513–527, MR 1119492. [18] P. Liu, Martingale and Geometry of Banach Spaces, Science Press, Beijing, 2007. [19] P. Liu, Y. Hou, Atomic decompositions of Banach-space-valued martingales, Sci. China Ser. A 42 (1) (1999) 38–47, MR 1692138. [20] P. Liu, L. Yu, B-valued martingale spaces with small index and atomic decompositions, Sci. China Ser. A 44 (11) (2001) 1361–1372, MR 1877222. [21] G. Lorentz, Some new functional spaces, Ann. of Math. (2) 51 (1950) 37–55, MR 0033449. [22] G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (3–4) (1975) 326–350, MR 0394135. [23] G. Pisier, Martingales in Banach Spaces, Cambridge University Press, 2016. [24] F. Weisz, Martingale Hardy spaces for 0 < p ≤ 1, Probab. Theory Related Fields 84 (3) (1990) 361–376, MR 1035662. [25] F. Weisz, Martingale Hardy Spaces and Their Applications in Fourier Analysis, Lecture Notes in Math., vol. 1568, SpringerVerlag, Berlin, 1994, MR 1320508. [26] F. Weisz, Dual spaces of multi-parameter martingale Hardy spaces, Q. J. Math. 67 (1) (2016) 137–145, MR 3471274. [27] L. Yu, Duals of Banach-space-valued martingale Hardy spaces, Kyungpook Math. J. 41 (2) (2001) 259–275, MR 1876197. [28] L. Yu, Duals of Hardy spaces of B-valued martingales for 0 < r ≤ 1, Acta Math. Sin. (Engl. Ser.) 30 (8) (2014) 1365–1380, MR 3229147.

1420

K. Liu et al. / J. Math. Anal. Appl. 450 (2017) 1401–1420

[29] L. Yu, Dual spaces of weak Orlicz–Hardy spaces for vector-valued martingales, J. Math. Anal. Appl. 437 (1) (2016) 71–89, MR 3451958. [30] L. Yu, Duality theorem for B-valued martingale Orlicz–Hardy spaces associated with concave functions, Math. Nachr. 289 (5–6) (2016) 756–774, MR 3486155.