An Equivalent Quasinorm for the Lipschitz Space of Noncommutative Martingales

Open Mathematics 2020; 18: 1281–1291

Research Article

Congbian Ma* and Yanbo Ren An equivalent quasinorm for the Lipschitz space of noncommutative martingales

https://doi.org/10.1515/math-2020-0072 received December 4, 2019; accepted August 10, 2020

Abstract: In this paper, an equivalent quasinorm for the Lipschitz space of noncommutative martingales is presented. As an application, we obtain the duality theorem between the noncommutative martingale Hardy space hc (resp. hr ) and the Lipschitz space λc (resp. λr ) for 0 p 1, β 1 1. p( ) p( ) β( ) β( ) << =−p c r We also prove some equivalent quasinorms for hp( ) and hp( ) for p = 1 or 2 <<∞p . Keywords: noncommutative space, martingale, Hardy space

MSC 2020: 46L53, 46L52, 60G42

1 Introduction

In the past two decades, due to the excellent work of Pisier and Xu on noncommutative martingale inequalities [1], the study of noncommutative martingale theory has attracted more and more attention. Especially in recent years, some meaningful research results on the noncommutative martingale theory have emerged continuously, and it has become a research hotspot in the field of noncommutative analysis. The Lipschitz space was first introduced in the classical martingale theory by Herz and plays an important role in it. For instance, the Lipschitz space is the generalization of the BMO space and the dual ( ) c r space of the Hardy space hp 0 <≤p 1 . The noncommutative Lipschitz spaces λ0( ) and λ0( ) were first introduced in [2], and with their help, the atomic decomposition for the Hardy space h1() was c r proved. In this paper, we study the noncommutative Lipschitz spaces λβ( ) and λβ( ) for β ≥ 0.We show that for q 0, β 1 , we have − ∞< < =−q c c r r λXβ()=()q and λXβ ()=(q ) with equivalent norms. As its application, we have the duality equalities for 0 p 1 and β 1 1 << =−p c ∗∗c r r (hλp())=()β and (())=() hλp β . This answers positively a question asked in [2]. The other main result of this paper concerns the equivalent c r quasinorms for hp( ) and hp( ) for p = 1 or 2 <<∞p . We prove the equalities c 2,c r 2,r hL1 ()=()1 and hL1 ()=()1 , c r which give a new characterization of h1 ( ) and h1 ( ).

 * Corresponding author: Congbian Ma, School of Mathematics and Information Science, Xinxiang University, Henan, People’s Republic of China, e-mail: [email protected] Yanbo Ren: School of Mathematics and Statistics, Henan University of Science and Technology, Henan, People’s Republic of China, e-mail: [email protected]

This paper is organized as follows. Some deﬁnitions and notations are given in Section 2.An equivalent quasinorm for noncommutative martingale Lipschitz space is shown in Section 3. Equivalent c r quasinorms for hp( ) and hp( ) for p = 1 or 2 <<∞p are considered in Section 4.

2 Preliminaries

Let be a von Neumann algebra acting on a Hilbert space H and τ be a normal faithful ﬁnite trace on . We call () , τ a noncommutative probability space. Let x be a positive operator on H. Then x admits a ∞ unique spectral decomposition: x =()∫ λexd λ . We will often use the spectral projection e(∞)λ, (x) 0 corresponding to the interval (λ, ∞). For 0 <≤∞p , let Lp() be the associated noncommutative Lp-space. Recall that the norm on Lp() is deﬁned by

p 1 ∥∥xτxxLp =((||))p ,, ∈p ( )

∗ 1 where |xxx|=( )2 is the usual modulus of x. Note that if p =∞, L∞() is just with the usual operator norm. For more detailed discussions about noncommutative Banach function spaces, see [3–5].

Let us recall the general setup for noncommutative martingales. Let ( nn) ≥1 be an increasing ﬁltration of von Neumann subalgebras of such that the union of n’s is weak*-dense in and n (with 0 = 1) the conditional expectation with respect to n. A sequence x =(xnn )≥1 is said to be adapted if xnn∈(L1 ) for all n ≥ 1, and predictable if xnn∈(L11 − ) for n ≥ 2. A noncommutative martingale with respect to the ﬁltration ( nn) ≥1 is a sequence x =(xnn )≥1 in L1() such that

nn()=xx+1 n for all n ≥ 1.

If additionally, x =(xLnn )≥1 ⊂ p () for some 0 ≤<∞p , we call x an Lp()-martingale. In this case, we set ∥∥xxpnn =sup ∥ ∥p.If∥∥x p <∞, then x is called a bounded Lp()-martingale. Note that the space of all bounded Lp-martingales, equipped with ∥⋅∥p, is isometric to Lp() for p > 1. This permits us to not distinguish a martingale and its final value x∞ (if the latter exists). Let x =(xnn )≥1 be a noncommutative martingale with respect to ( nn) ≥1 with the usual convention that 0 = 1.Define dxxxnnn=−−1 for n ≥ 1 with the usual convention that x0 = 0. The sequence ddxx=(nn )≥1 is called the martingale difference sequence of x. In the sequel, for any operator x ∈(L1 ) we denote xnn=( x) for n ≥ 1. Let x =(xnn )≥1 be a finite martingale in L2(). We set

1 1  n  2  ∞  2 2 2 scn, ()=xxsxx∑∑kk−1 (|d, |) c ()= nn−1 (| d |) k=1  n=1  and

1 1  n  2  ∞  2 ∗ 2 ∗ 2 srn, ()=xxsxx∑∑k−1 (|d,k |) r ()= n−1 (| dn |) . k=1  n=1  These will be called the column and row conditioned square functions, respectively. Let 0 <<∞p .Define c r ( ) fi - ( ) c hp( ) resp. hp( ) as the completion of all nite L∞ martingales under the quasi norm ∥∥x hp() = c r ( r ) fi ( ) ∥()∥sxc p resp. ∥∥xsxhrp() =∥ ()∥p . For p =∞.Wede ne h∞( ) resp. h∞( ) as the Banach space of the - ∞ 2 ( ∞ ∗ 2) L∞ martingales x such that ∑n=1 nn−1(|dx |) respectively ∑n=1 n−1(|dxn |) converge for the weak operator topology. For more information of noncommutative martingales, see the seminal article of Pisier and Xu [3] and the sequels to it. c r The main object of this paper is the noncommutative Lipschitz spaces λβ and λβ( ). An equivalent quasinorm for the Lipschitz space of noncommutative martingales  1283

Deﬁnition 2.1. Let β ≥ 0. Set c c λxLxβ()={∈()∥∥2 : λβ() <∞} with

 β 1 β 2 1  c − −−2 2 ∥∥xτexτeτexxλβ() =max sup () ∥1 ()∥∞ , sup sup () (nn (|− |)) , e∈ 1 ne≥∈1 n  where n denotes the lattice of projections of n. Similarly, we deﬁne

r r ∗ c λxLxxβ( )={ ∈2 ( ): ∥ ∥λλβ() =∥ ∥β() <∞}. 2 ﬁ [ ] The classical martingale space L1 which is de ned in 6 has the following noncommutative analogue.

Deﬁnition 2.2. We deﬁne

∞  2,c 1 LxLxyzyLzLz()=∈():, =∑ k ∈(22 mk ) ,,0 ∈() mk ()= 1  k k kk  k=1 ∞    and∥∥xyzL 2,c() = inf∑ ∥k ∥∥22k ∥ <∞ , 1    k=1   ﬁ where ( mkk) ≥1 and ( mkk) ≥1 are the subsequences of ( kk) ≥1 and ( kk) ≥1, the in mum runs over all decompositions of x as above. Similarly, deﬁne

2,r ∗ L xL: x2,rc x2, . 1 ( )={ ∈1 ( ) ∥ ∥LL1 () =∥ ∥1 () <∞} 2,c 2,r It is clear that L1 ( ) and L1 ( ) are Banach spaces. We will use the following deﬁnitions from [2,7].

Deﬁnition 2.3. [2] Let 02<≤p . Denote the index class W1 which consists of sequences (ωnn) ≥1 such that 21/−p + ﬁ (ωn )n≥1 is nondecreasing with each ωnn∈(L1 ) invertible with bounded inverse and ∥∥≤ωn 1 1.De ne

1  2     ∞   c c 12−/p 2 Xp()=∈()∥∥xL2 :infd xX () = τ∑ || ωn | xn+1 | <∞. p W     1 n=1      

12−/p For 2 <≤∞p or − ∞≤p <0. Denote W2 which consists of sequences (ωnn) ≥1 such that (ωn )n≥1 is + nondecreasing with each ωnn∈(L1 ) invertible with bounded inverse and ∥∥≤ωn 1 1.Deﬁne

1    ∞  2  c  12p 2  X ()=∈()∥∥xL:supd xc = τ || ω−/ | x | <∞. p  2 Xp()  ∑ n n+1    W2 n=1       Similarly, deﬁne

r r ∗ c Xp( )={xL ∈2 ( ): ∥ x ∥XXp() =∥ x ∥p() <∞}. c Remark that for 2 <≤∞p , Xp( ) can be rewritten in the following form. Given (ωWnn)∈≥1 2, we set

1 2 2 2 gωωα α ,1, n n =−()n n−1 ∀≥ where α 1 1 . It is clear that =−2 p

α   ∞  2     2   ( ) (gGn)∈αnnnαn = uu =( )≥0 :, uL ∈ ( ) τ∑ | un | ≤ 1. 3.1      n=1    1284  Congbian Ma and Yanbo Ren

Then

1   ∞  2 22 ∥∥xτuxxc =sup  | | (|− |) . Xp()  ∑ nn n ()∈uGnα n=1 

We recall the deﬁnition of the space Lp(ℓ ; ∞), with 1 ≤≤∞p . A sequence (xnn) ≥1 in Lp() belongs to Lp(ℓ; ∞) if (xnn) ≥1 admits a factorization xn = ayn b, with a, bL∈(2p ) and (yLn)∈(n≥∞1 ). The norm of (xnn) ≥1 is then deﬁned as

∥()xaybnn≥(ℓ)1; ∥ Lp ∞ =inf {∥∥ 2p sup ∥n ∥∞ ∥∥2p }. xaybn= n n≥1

+ We usually write ∥(xxnn )≥(ℓ)1; ∥ Lp ∞ = ∥sup n ∥p. n Deﬁnition 2.4. [7] Let 2 <<∞p .Deﬁne the space

c + 2 p Lp mo()=∈()∥ a L2 :sup nn (|−()|)∥<∞ a a { 2 } equipped with the norm

1  2  c +2 p ∥aaaa ∥Lmo() =max ∥ ∥ p , ∥ sup n (| − n ( )| )∥ . p  ()2 

c c Then (Lmop ()∥⋅∥, Lmop ()) is a Banach space. Similarly, we set r ∗ c Lp mo()={ a: a ∈ Lp mo ()} equipped with the norm

r ∗ c ∥∥aaLmop () =∥ ∥ Lmop ().

3 An equivalent quasinorm for the Lipschitz space of noncommutative martingales

c In this section, we prove the noncommutative equivalent quasinorms for Lipschitz spaces λβ( ) and r ( ) c ( r ) λβ( ) β ≥ 0 . As its application, we obtain the duality of the Hardy space hp( ) resp. hp( ) and the c ( r ) Lipschitz space λβ( ) resp. λβ( ) for 0 <

Theorem 3.1. For q 0 and β 1 , we have λXc c with equivalent norms. More precisely, − ∞< < =−q β()=(q ) for any x ∈(L2 ), we have

− 1  2  2 1 xxxc c c .  −∥∥≤∥∥≤∥∥ XλXq ()β ()q ()  q  r r Similarly, λXβ()=(q ) with the same equivalence constants.

The following Lemma is the key ingredient of our proof.

Lemma 3.2. c For x ∈(λβ ) (β ≥ 0), we have   22 2β ( ) nn(|xx − | ) ≤ C ∑ τeen ( ) ≥ 1, 3.1   e∈ n 

c where C =∥x ∥λβ( ) and n is the set consisting of all minimal projections with respect to n. An equivalent quasinorm for the Lipschitz space of noncommutative martingales  1285

Proof. ﬁ c By the de nition of λββ()(≥0), we have for any e ∈ n

2 1 1 +β ( ) τ((|−|))≤()exxnn2 Cτe2 . 3.2

Let P be a projection with respect to n and there does not exist any minimal projections e′ such that 2 e′≤P. Let y =(|−|)nnxx P. Then for any k ≥ 1 1 yey11,,∞∞()≥ey (). ()kkk () Thus by (3.2), we get that

1 1 1 1 +β 1 2 2 2 2 2 τeyτyeyτxxPeyCτeyP1 ,,∞∞()≤1 ()=nn (|−|)1 ,, ∞ ()≤1 ∞ () . k ()()(()kk () () k )() () k

When τ(())≠ey1 ,∞ 0, we have that ()k β 1 ( ) ≤()τe1 ,∞ y . 3.3 k ()()k

ﬁ Note that e 1 ,∞ ()≤yPand P can be divided into in nite small pieces. Thus, τ(())=ey1 ,∞ 0, which ()k ()k ( ) 2 contradicts 3.3 . Therefore, we obtain τ(())=ey1 ,∞ 0 for every k ≥ 1 which implies nn(|xx − | ) P = 0. ()k

( ) β 2 2 Now we prove 3.1 holds. Let e ∈ n and a =()Cτ e . Let ε > 0 and e0 =((|−|)exx((aε + ), ∞) nn). Then we have that

22 eexxaεee0nn(| − | ) ≥ ( + ) 0 . Thus using (3.2), we get that

1 2 1 1 (a+)( ετe00 e )≤(2 τe eΕnn (|− x x |))≤2 aτee (0 )2 .

It is easy to see that e0e = 0, which implies that e ≤ e0 does not hold. Using the preceding result, we have that

2 nn(|xx − | ) e0 = 0.

2 2 Let e1 =((|−|)exx()0,a nn). Then we have 222 nn(|xx − | ) = nn (| xx − | ) e1 ≤ ae1. It follows that

222β ( ) nn(|xx − | ) eCτee ≤ ( ) . 3.4 Note that (|xx − |2 )(1 − ∑ e ) = 0. Thus by (3.4), we have nn e∈n   2222β nn(|xx − | ) = nn (| xx − | )∑∑ eC ≤ τee ( ) .   ee∈∈nn  The proof is complete. □

We will also need the following well-known lemma from [8].

Lemma 3.3. Let f be a function in 1(+) and x, y ∈ +. Then

 1  τ((fx + y )−())= fx τ f ′(+ x tyy )d t. ∫   0  1286  Congbian Ma and Yanbo Ren

Proof of Theorem 3.1. Let x X c and 111 . Fix an integer m ≥ 0 and let e be a projection with ∈(q ) αq=−2 respect to m.Deﬁne

1 eτ() e− α ,; m = n un =  0,mn≠ .

Noting that (uGnn)∈≥0 α, we have

1 1   ∞  2 2 2 2 22 τe()(− α τe |−|)= x x τ ||(|−|)≤∥∥ u x x x c . ( nn)  ∑ nn n Xq ()  n=1 

c c c Thus, we have that x ∈(λβ ) and ∥∥xxλXβ() ≤∥∥q (). c c Now, let x ∈(λβ ) and C =∥x ∥λβ( ). Let n be the set consisting of all minimal projections with respect to n. Then by Lemma 3.3, we have that   22 2β ( ) nn(|xx − | ) ≤ C ∑ τee ( ) . 3.5   e∈ n  α 2 2 Let u =(uGn )∈ α. Denote run′ =∑ |k | and rrern = nn′ (∞)0, ( ′). Then rn is invertible and τ()≤rn 1. Let ∞ ()kn≤ ∞ rλen = ∫ d λ be the spectral decomposition of rn. Let ddμτe=(λ). Then we have ∫ λμd ≤ 1. Observe that 0 0 μλ χ λ λ 1, ∑ ()0 {}λ0 ()⋅≤ λ0∈(0, ∞) where χλis the characteristic function at point λ . It follows that {}λ0 ( ) 0 μλ22β χ λ λ− β. ∑ ()0 {}λ0 ()≤ λ0∈(0, ∞) By the continuous function calculus, we have

2β −2β ∑ τe() e ≤ rn . (3.6) e∈n Using (3.5) and (3.6),

1 1 1 ∞ 2 ∞ 2 ∞ 2         2 2 2  τ|| u22 (|− x x |)≤ Cτ  || u22 τe ()β e ≤ Cτ ( rα − rα ) r1− α  . (3.7)  ∑∑∑∑nn n n   n n−1 n      n=1   n=1 e∈=n   n 1 

α 2 2 α α Applying Lemma 3.3 with f ()=ttxyr2 , + = n and x = rn−1, we obtain

1  α  2 2 2 2 2 2 2 −1 2 2 α 1− α α α 2 τ rrrα −≤+−−=(−)α τ r trrα rrα d t τrnn r 1 , ()()()n n−1 n ∫ n−−11 ()()n n n n−1  α −  0 

α −1 α where we have used the fact that the operator function a ↦ a 2 is nonincreasing for − 11<−≤0. 1 2 1 2  ∞ 222  2  Taking the sum over n leads to τuxx∑||(|−|)≤nn n   ∥∥ xλ c(). Taking supremum over all  ()n=1  α  β (uGn)∈ α, we get

1 1  2  2  2  2 x c   x 1 x . ∥∥Xλp() ≤ ∥∥ββ () = − ∥∥ λ ()  α   p The proof is complete. □

Using the dual result in Theorem 3.2 in [2], we will describe the dual space of hpp()(<< 01) as the Lipschitz space. An equivalent quasinorm for the Lipschitz space of noncommutative martingales  1287

Corollary 3.4. Let 0 p 1 and β 1 1. Then we have << =−p c ∗ c (hλp())=(β ) and r ⁎ r (hλp())=(β ) with equivalent norms.

c c ( ) 4 Equivalent quasinorms for h1 ( ) and hp( ) 2 <

ﬁ c In this section, we rst describe an equivalent quasinorm for h1 ( ). As in the classical case, the spaces h1 2 ( [ ]) and L1 are equivalent see 6 . We will transfer this to the noncommutative martingales.

Theorem 4.1. We have that c 2,c r 2,r hL1 ()=()1 andhL1 ()=(1 ) with equivalent norms.

For the proof we need the following lemmas.

Lemma 4.2. Let 1 ≤≤p 2 and p′ be the conjugate index of p. For x, yL∈(2 ), we have

1  2  2 τxy∗ xyc c . | ()|≤  ∥∥∥∥hXp()p′ ()  p

Proof. Let (rWn)∈ 1. Then by the Cauchy-Schwarz inequality and Lemma 3.1, we have that

1 1 ∞ ∗ ∞ 2 ∞ 2 1 1 1 − 1  2   1− 2  ∗  2 − pp 2 ′′  1− p2  p2  τxy()= τdd xr yr ≤ τ r | d x | τ r | d y | ∑∑∑()k k−11k k−   ()k−1 k   k−1 k  k=1     k=1  k=1   1  2  2 xyc c xyc c . ≤∥ ∥XXp() ∥ ∥p′′ () ≤  ∥ ∥ hXp () ∥ ∥ p ()  p

Note that the set Gα deﬁned in (3.1) can be reduced to the following one:

α   2       GuuuLτufinite sequences : , 2  1 . αnnnn==()∈()||≤ ≥∞1 ∑ n     n 1     ≥   

Indeed, for any (uGnn)∈≥0 α set

()N ||uenNn[]0, (||)≤ u,; n N un =  0,nN> . Then we have that

1 1    2    2 ()N 2222 lim τu∑∑|n |nn (| xx − | ) = τu |nn | (| xx − n | ) . N→∞        n≥1   n≥1 

Thus, the set Gα can be reduced to Gα. □ 1288  Congbian Ma and Yanbo Ren

Lemma 4.3. Let y ∈(L2 ) and 2 ≤≤∞p . Then we have

c ∗ ∥∥yτxyXp() =sup |( )|, p xF∈ 2 where

1  2       F p ==xabaGb:,0,1 ()∈()=  τb ||≤2  . 2  ∑∑nn n α n n   n    n≥≥11n      

Proof. For any ﬁnite sequences (aGn)∈α, set

1    2 Aτay=|||| 22n  , ay,  ∑ n   n≥1 

n where y =−yyn. Then we have Aay, <∞and

A 2  yan ∗   ay, n∗ n n∗ ( ) Aay, ==τay∑∑n  =τayz n n, 4.1 A  A    ay, n≥1 ay,  n≥1  n ∗ yan 1 n ∗ where zn = . Note that nn()=zyan ( )n =0 and Aay, Aay,       2 11n ∗ 2 22n τ ∑∑||zn  =τya |n | =τay ∑ ||||n  =1.   A 2   A 2   n≥≥1  ay, n 1  ay, n≥1 

p Let x =∑n≥1 aznn. It is clear that x ∈ F2 and

∑∑aznn ≤∥∥∥∥≤∥∥∥∥<∞ ann z2 sup an ∑ zn 2 . n 1 n≥≥1 2 n 1 ≥ n≥1 Thus,

  τazyn∗ az yy∗ ∗ . ∑∑∑nn  ≤−<∞nn n n≥11 n≥≥2 n 1 2 ﬁ c ( ) Therefore, by the de nition of Xp( ) and 4.1 , we have that

  n ∗ c sup sup∗ sup ∗ ∥yAτazyτazyy ∥Xp() =ay, =∑∑nn  = (nnn ( ( −n )))  pp  ()∈aGnα xF ∈22n≥11 xF∈ n≥

∗ ∗ ∗ =()−(())=|()|sup∑∑τaznn y τannn z y sup τxy . □ p n p xF∈ 2 n≥11n≥ xF∈ 2

Proof of Theorem 4.1. First let x = zy and there exists n ≥ 1 such that y ∈(L2 n), zL∈(2 ), n()=z 0. Then we have that

mmn()=zy ()= zy0; m ≤ n. (4.2) Thus, we have that

mm()=()zy z y;1 m ≥. (4.3)

Using (4.2), (4.3), and the fact y ∈(L2 n),weﬁnd

  2 2 2 ∗ 2 sc()xzyzyyzy =∑∑kk−1 |dd | =kk−1 | | = ∑kk−1 | d | . k≥1 kn> kn>  An equivalent quasinorm for the Lipschitz space of noncommutative martingales  1289

Thus, we deduce that

c ∥xsxszyszyzy ∥hc1 () =∥ ( )∥=∥1 c ( ) ∥ 1 ≤∥ c ( )∥∥ 22 ∥ =∥∥∥ 22 ∥.

Therefore, we get that xxc 2,c . Now we consider the general case. Let ∥∥h1 () ≤∥∥L1 ( ) ∞ x = ∑ zykk, k=1 k k k ( ) where for every k ≥ 1, y ∈(L2 mk), zL∈(2 ), and mk()=z 0. Then by 4.3 , we have that

n n ∞ kk kk kk c c ∥∥xzyzyzyh1 () =lim∑∑∑ ≤ lim ∥ ∥h1 () ≤ ∥ ∥∥22 ∥. n→∞ n→∞ k=111c k= k= h1 ()

It follows that xxc 2,c . ∥∥h1 () ≤∥∥L1 ( ) 2,c We turn to the converse inequality. Let x ∈(L1 ). Then by Lemma 4.2 and Lemma 4.3,

∗ xτxy2,c sup ∥∥L1 () = |( )| 2,c ∗ yL∈()∥∥1 ,1 y 2,c ∗ ≤ L1 ()

sup xycc ≤∥∥∥∥hX1 ()∞ () (4.4) 2,c ∗ yL∈()∥∥1 ,1 y 2,c ∗ ≤ L1 () c ∗ ≤∥xτay ∥h1 () sup sup | ( )| . 2,c ∗ p yL∈()∥∥1 ,1 y2,c ∗ ≤∈ aF2 L1 ()

We will show sup|(τay∗ )|≤ sup |( τay∗ )|. p 2,c (4.5) aF∈ 2 aL∈()∥∥1 ,1 a 2,c ≤ L1 ()

p Indeed, let a ∈ F2 . Then a can be decomposed as

a = ∑ rbnn, (4.6) n≥1

1  2 2 - where (rGnαnn)∈, ( b )=0 and (τb(∑n≥1 |n | )) ≤ 1. Thus, by the Cauchy Schwarz inequality we have that

1 1   2   2 rb r2 b2 1. ∑∑∑∥∥∥∥≤nn22 ∥∥n 2   ∥n ∥2  ≤ n≥≥≥1 n 1  n 1 

2,c Taking the inﬁmum as in (4.6), we obtain a L and a 2,c 1, which imply that (4.5) holds. ∈(1 ) ∥∥L1 () ≤ c 2,c Combining (4.4) and (4.5), we have that xx2,c c . Thus, we have that hL. ∥∥L1 () ≤∥∥h1 ( ) 1 ()=(1 ) r 2,r □ Similarly, we have that hL1 ()=(1 ). The proof of the theorem is complete. c Let 2 <<∞p . Our second result of this section concerns the equivalent quasinorms for hp( ) r and hp( ).

Theorem 4.4. Let 2 <<∞p . Then we have c c c hXLmop()=()=p p () with equivalent norms. More precisely,

1  p 2 ∥∥xxXLmohc() ≤∥∥c () ≤∥∥ xc () ≤  ∥∥x Xc () . p p p  2  p Similarly, r r r hXLmop()=()=p p () with equivalent norms. 1290  Congbian Ma and Yanbo Ren

Proof. c Step 1: Let x ∈(Xp ). Fix a positive integer N, we will show that scN, ()∈xL p ( ). Let 1 ≤≤nN. Since the dual space of L p( n) is L p ( n), 2 p−2

2 p 2 ∥(|nndsupdxτyx+1 |)∥= (n nn (|+1 |)). 2 + yLn∈()p n p−2

∥∥yn L p () ≤1 p−2 p p−2 Let wn = yn . Then wn ≥ 0 and τ()≤wn 1. Thus, we have that

1− 2 2 p p 2 2 ∥(||)∥=nndsupdxτwxx+1 n nn (||)≤∥∥+1 c . 2 ()Xp() wτwnn≥()≤0, 1 Therefore, we obtain that

1 1 N 2 N 2 N   1 2 2 2 2 p c ∑∑∑nn−1(||)=dddxxxNx nn−1 (||)≤ ∥nn−()1 (||)∥≤∥∥<∞ Xp .   2 n=1  n=1 p n=1 p 2

p p p Assume ∥()∥=sxcN, p 1. Then the sequence {sxsxcN,,(),,,,cN ()… sxcN , ()…}∈ W2. Set sc,0()=x 0. Thus, we have that

N N  2    2 p 1− p 2 2 p−2 2 2 p ∥∥xc ≥ τ ( sx, ()) ( sxscn, ()− cn,1− ())= x τsx , () ( sxscn, ()− cn,1− ())= x τsx (, ()). Xp() ∑∑cN   cN  cN n=1   n=1  It follows that

c c ∥∥xxhXp() ≤∥∥p (). c Now let x ∈(hp ) and (wWn)∈ 2. Set sc,0()=x 0. Then for any n

1 n 2 1   1− 2  1− 2 2 21 pp2 2  2  1− ppp τ∑ wk ( sck,1+ ()− x s ck, ()) x = τ wn ( scn,1+ () x ≤(( τwn )) (( τscn,1+ ))≤∥ scp ()∥ x .     ()  k=1  Thus, we get that

1 ∞ 2   1− 2  τwsxsxp(()−())≤∥()∥2 2  sx.  ∑ k ck,1+ ck,  cp  k=1 

c c Therefore, the inequality ∥∥xxXhp() ≤∥∥p () holds. c Step 2: Let x ∈(hp ). Then for any n ≥ 0  ∞  2 2 2 nn(|xx − ( )| ) = n ∑ kk−1 (|d x | ) ≤ n ( sxc ( )). kn=+1  Thus, we have that

c c ∥∥xxLmop () ≤∥∥ hp ().

Now let x ∈()Lmoc ) and 11+=1. Note that p pp′   + 2  2  ∥((|−()|))∥=supnnxxp sup τxxyyL ((|−()|))∈ nn :p′ and y ≤ 1. 2 ∑∑nn 2 n n≥≥1 n 1 p′   2  Thus, we have that An equivalent quasinorm for the Lipschitz space of noncommutative martingales  1291

1   ∞  2 c 22+ 2p c ∥∥xτuxxxxxX () =sup  |nn | (|− n | ≤∥ sup ( n (|− n ()|))∥≤∥∥ Lmo() . p  ∑  2 p ()∈uGnα n=1  The proof of the theorem is complete. □

The following is an immediate consequence of Theorem 4.4 and Theorem 3.3 in [2](or Theorem 3.1 in [7]).

Corollary 4.5. Let 1 <

r ∗ r (hhp())=(q ) with equivalent norms.

Acknowledgment: This work was supported by the National Natural Science Foundation of China (11871195, 11671308, and 11471251).

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