PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Number 7, July 2010, Pages 2431–2441 S 0002-9939(10)10267-6 Article electronically published on March 24, 2010 BURKHOLDER’S INEQUALITIES IN NONCOMMUTATIVE LORENTZ SPACES YONG JIAO (Communicated by Marius Junge) Abstract. We prove Burkholder’s inequalities in noncommutative Lorentz spaces Lp,q(M), 1 <p<∞, 1 ≤ q<∞, associated with a von Neumann algebra M equipped with a faithful normal tracial state. These estimates generalize the classical inequalities in the commutative case. 1. Introduction Martingale inequalities and sums of independent random variables are important tools in classical harmonic analysis. A fundamental result due to Burkholder [1, 2] can be stated as follows. Given a probability space (Ω, F ,P), let {Fn}n≥1 be a nondecreasing sequence of σ-fields of F such that F = ∨Fn and let En be the conditional expectation operator relative to Fn. Given 2 ≤ p<∞ and an p L -bounded martingale f =(fn)n≥1, we have ∞ ∞ 1/2 1/p 2 p (1.1) fLp ≈ Ek−1(|df k| ) + |df k| . Lp Lp k=1 k=1 The first term on the right is called the conditioned square function of f, while the second is called the p-variation of f. Rosenthal’s inequalities [14] can be regarded as the particular case where the sequence df =(df 1,df2, ...) is a family of independent mean-zero random variables df k = ak. In this case it is easy to reduce Rosenthal’s inequalities to ∞ ∞ ∞ 1/2 1/p ≈ 2 p (1.2) ak Lp ak 2 + ak p . k=1 k=1 k=1 Noncommutative analogues of the above inequalities were successfully obtained by Junge and Xu in [10] and [11]. They replaced conditioned expectations onto the σ- subfields by conditioned expectations onto an increasing sequence of von Neumann subalgebras of a given von Neumann algebra. More precisely, for 2 ≤ p<∞ and Received by the editors January 13, 2009, and, in revised form, September 22, 2009. 2000 Mathematics Subject Classification. Primary 46L53; Secondary 60G42. Key words and phrases. Noncommutative martingales, Burkholder’s inequalities, Lorentz spaces. The author was partially supported by the Agence Nationale de Recherche (06-BLAN-0015), the National Natural Science Foundation of China (10671147) and the China Scholarship Council (2007U13085). c 2010 American Mathematical Society Reverts to public domain 28 years from publication 2431 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2432 YONG JIAO p any finite noncommutative L (M)-martingale x =(xn)n≥1, (1.1) has the following noncommutative version: 1/p p p M ≈ | | (1.3) x L ( ) max dxk Lp(M), sc(x) Lp(M), sr(x) Lp(M) , k where sc(x)andsr(x) denote column and row versions of the conditioned square function. Moreover, they obtained a simpler inequality for 1 <p≤ 2 by duality. Recently, Randrianantoanina [15] proved a weak-type inequality for conditioned square functions, which implies Junge and Xu’s noncommutative Burkholder in- equalities by interpolation. This alternate approach yields better constants, some of which are optimal. Our original motivation comes from the classical extension to Lorentz spaces of Rosenthal’s inequalities (1.2) by Carothers and Dilworth [5], i.e., for 2 <p< ∞, 0 <q≤∞, and any independent mean-zero random variables f1,f2, ..., fn, n n n p,q ≈ ⊕ (1.4) fk L (Ω) max fk L2(Ω), fk Lp,q (0,∞) , k=1 k=1 k=1 n ⊕ where k=1 fk denotes the disjoint sum of f1,f2, ..., fn, which is a function on ∞ n (0, ) with df (t)= k=1 dfk (t). Inspired by (1.3) and (1.4), in this paper we consider Burkholder’s inequalities in noncommutative Lorentz spaces Lp,q(M), 1 <p<∞, 1 ≤ q<∞, and one of our main results can be stated as follows (see Theorem 3.1 for the detailed statement): for 2 <p<∞, 1 ≤ q<∞, and any finite Lp,q(M)-martingale x, we have (1.5) p,q M ≈ ⊗ x L ( ) max dxk ek Lp,q (M⊗∞), sc(x) Lp,q (M), sr(x) Lp,q (M) . k Note that if p = q, we come back to the inequalities (1.3). We also extend these inequalities to the case where 1 <p<2, 1 ≤ q<∞. Our main results are contained in section 3. Note that the proofs of these inequalities for Lp-spaces in [10] and [11] use an iteration argument; however, this iteration seems inefficient (or more complicated) for the case of Lorentz spaces. We will adopt a different approach based on the Randrianantoanina weak-type (1,1) inequality. 2. Preliminaries Let (Ω, F ,P) be a (commutative) probability space and f a random variable on (Ω, F ,P). The decreasing rearrangement of f, denoted by f ∗, is f ∗(t)=inf{s>0: P (|f| >s) ≤ t},t ∈ [0, 1]. The Lorentz space Lp,q(Ω) = Lp,q, 0 <p<∞, 0 <q≤ ∞, consists of those measurable functions with finite quasi-norm fp,q given by ∞ 1/q 1/p ∗ q dt fp,q = t f (t) , 0 <q<∞, 0 t 1/p ∗ fp,∞ =supt f (t),q= ∞. t>0 It is well known that if 1 <p<∞ and 1 ≤ q ≤∞, then fp,q is equivalent to a norm. H¨older’s inequality for Lorentz spaces is the following, which first appeared in the work of O’Neil [8]: ≤ fg p,q C f p0,q0 g p1,q1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BURKHOLDER’S INEQUALITIES IN NONCOMMUTATIVE LORENTZ SPACES 2433 for all 0 <p,q,p0,q0,p1,q1 ≤∞such that 1/p =1/p0 +1/p1 and 1/q =1/q0 +1/q1. We refer to [3], [4], and [7] for details on classical (commutative) Lorentz spaces. Now we introduce the noncommutative Lorentz spaces. Let (M,τ) be a tracial noncommutative probability space. Namely, M is a von Neumann algebra with a normal faithful normalized trace τ. We refer to [13] for noncommutative integration and additional historical references. We only briefly recall some elementary facts about noncommutative Lorentz spaces. Let L0(M) denote the topological ∗-algebra of all measurable operators with respect to (M,τ). For x ∈ L0(M), define its generalized singular number by μt(x) = inf{λ>0: τ 1(λ,∞)(|x|) ≤ t},t>0. Then for 0 <p<∞, p p L (M)={x ∈ L0(M):τ(|x| ) < ∞} and ∞ p | |p p x Lp(M) = τ( x )= μt(t) dt. 0 Of special interest in this paper are the noncommutative Lorentz spaces Lp,q(M) associated with (M,τ): p,q L (M)={x ∈ L0(M):xLp,q (M) < ∞}, where ∞ 1/q 1/p q dt xLp,q (M) = t μt(x) 0 t for 0 <q<∞ and with the usual modification for q = ∞. The noncommutative Lorentz spaces behave well with respect to real interpola- tion. Let 0 <θ<1, 0 <pk,qk ≤∞for k =0, 1andp0 = p1.Then p,q p0,q0 p1,q1 L (M)=[L (M),L (M)]θ,q, where 1/p =(1− θ)/p0 + θ/p1, 0 <q≤∞. The usual H¨older inequality also extends to the noncommutative setting. Let 0 <pk,qk ≤∞for k =0, 1and1/p =1/p0 +1/p1, 1/q =1/q0 +1/q1. Then for any x ∈ Lp0,q0 (M)andy ∈ Lp1,q1 (M), ≤ (2.1) xy Lp,q (M) C x Lp0,q0 (M) y Lp1,q1 (M). In particular, if p = q =1, | |≤ ≤ ∀ ∈ p0,q0 M ∈ p1,q1 M τ(xy) xy L1(M) x Lp0,q0 (M) y Lp1,q1 (M), x L ( ),y L ( ). For 1 <p<∞, 1 ≤ q<∞, this defines a natural duality ∗ Lp,q(M) = Lp ,q (M), where p,q denote the conjugate indices of p, q, respectively, and x, y = τ(xy). Let (Mn)n≥1 be an increasing sequence of von Neumann subalgebras of M such ∗ that the union of the Mn is weak -dense in M. For each n ≥ 1, it is well known that there is unique normal faithful conditional expectation En from M onto Mn. p,q p,q Moreover, En extends to a bounded projection from L (M)ontoL (Mn)for 1 <p<∞, 1 ≤ q ≤∞, which we still denote by En. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2434 YONG JIAO For 1 ≤ p<∞, 1 ≤ q ≤∞, and a finite sequence a =(an)n≥1 in M, we define 1/2 2 aLp,q (M;2) = |an| , c Lp,q (M) n 1/2 | ∗ |2 a Lp,q (M;2) = an r Lp,q (M) n and 1/2 2 p,q 2 E | | a L (M,En−1; ) = n−1 an , c Lp,q (M) n 1/2 ∗ 2 p,q 2 E | | a L (M,En−1; ) = n−1 an . r Lp,q (M) n p,q Now, any finite sequence a =(an)inL (M) can be regarded as an element in p,q 2 L (M⊗B( )). Therefore, · p,q M 2 defines a quasi-norm on the family of all L ( ,c ) finite sequences in Lp,q(M). The corresponding completion is a quasi-Banach space, p,q M 2 ∞ denoted by L ( ,c )(ifq = , the completion should be taken in a certain weak topology). It is shown in [9] that · p M E 2 is a quasi-norm. Similarly, we L ( , n−1;c) can show that · p,q M E 2 defines a quasi-norm on the family of all finite L ( , n−1;c) sequences in Lp,q(M).