TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 367, Number 6, June 2015, Pages 4079–4110 S 0002-9947(2014)06185-0 Article electronically published on December 10, 2014
NONCOMMUTATIVE BOYD INTERPOLATION THEOREMS
SJOERD DIRKSEN
Abstract. We present a new, elementary proof of Boyd’s interpolation the- orem. Our approach naturally yields a noncommutative version of this re- sult and even allows for the interpolation of certain operators on 1-valued noncommutative symmetric spaces. By duality we may interpolate several well-known noncommutative maximal inequalities. In particular we obtain a version of Doob’s maximal inequality and the dual Doob inequality for non- commutative symmetric spaces. We apply our results to prove the Burkholder- Davis-Gundy and Burkholder-Rosenthal inequalities for noncommutative mar- tingales in these spaces.
1. Introduction Symmetric Banach function spaces play a pivotal role in many fields of mathe- matical analysis, especially probability theory, interpolation theory and harmonic analysis. A cornerstone result in the interpolation theory of these spaces is the Boyd interpolation theorem, named after D.W. Boyd. Together with the Calder´on- Mitjagin theorem, which characterizes the symmetric Banach function spaces which 1 ∞ are an exact interpolation space for the couple (L (R+),L (R+)), Boyd’s theorem provides an invaluable tool for the analysis of symmetric spaces. The history of Boyd’s interpolation theorem begins with the announcement of Marcinkiewicz [28], shortly before his death, of an extension of the Riesz-Thorin theorem. Let us say that a sublinear operator T is of Marcinkiewicz weak type (p, p) p if for any f ∈ L (R+), 1 −1 p ≤ p d(v; Tf) Cv f L (R+) (v>0), where d(·; Tf) denotes the distribution function of Tf. Marcinkiewicz demon- strated that if a sublinear operator T is simultaneously of Marcinkiewicz weak r types (p, p)and(q, q)for1≤ p
Received by the editors April 12, 2012 and, in revised form, March 12, 2013. 2010 Mathematics Subject Classification. Primary 46B70, 46L52, 46L53. Key words and phrases. Boyd interpolation theorem, noncommutative symmetric spaces, Φ-moment inequalities, Doob maximal inequality, Burkholder-Davis-Gundy inequalities, Burkholder-Rosenthal inequalities. This research was supported by VICI subsidy 639.033.604 of the Netherlands Organisation for Scientific Research (NWO) and the Hausdorff Center for Mathematics.
c 2014 American Mathematical Society Reverts to public domain 28 years from publication 4079
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4080 SJOERD DIRKSEN
The class of sublinear operators which are simultaneously of weak types (p, p)and (q, q) was subsequently characterized by Calder´on [6] as consisting of precisely those maps T which satisfy
μt(Tf) ≤ CSp,q(μ(f))(t)(t>0),
where μ(f) denotes the decreasing rearrangement of f and Sp,q is a linear integral operator which is nowadays known as Calder´on’s operator. Finally, in [5] Boyd introduced two indices pE and qE for any symmetric Banach function space E on R+ and showed that the operator Sp,q is bounded on E precisely when p
d(αv; Tf) ≤ d(v;Θp,qf)(v>0),
where Θp,q is the linear operator defined by
− 1 − 1 Θp,qf(s, t)=f(s)(t q χ(0,1)(t)+t p χ[1,∞)(t)).
Secondly, we show that Θp,q is bounded on E if p License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NONCOMMUTATIVE BOYD INTERPOLATION THEOREMS 4081 established in [7], as well as the Burkholder-Rosenthal inequalities for noncommu- tative Lp-spaces and Lorentz spaces obtained in [18] and [15], respectively. During the writing of this manuscript we discovered that an interpolation result for noncommutative Φ-moment inequalities associated with Orlicz functions was proved recently in [1]. We discuss the connection of our work with this result and in fact show that many of our interpolation results have a ‘Φ-moment version’. The paper is organized so that the first part, up to the classical Boyd interpola- tion theorem, can be read without any knowledge of noncommutative analysis. 2. Symmetric quasi-Banach function spaces In this preliminary section we introduce symmetric quasi-Banach function spaces and discuss their most important properties. The results presented below are all well known for Banach function spaces, but not easy to find for quasi-Banach func- tion spaces. We shall need the following well-known result due to T. Aoki and S. Rolewicz, which states that every quasi-normed vector space can be equipped with an equivalent r-norm (see e.g. [21] for a proof). Theorem 2.1 (Aoki-Rolewicz). Let X be a quasi-normed vector space. Then there is a C>0 and 0 n n 1 r ≤ r (1) xi C xi X . X i=1 i=1 Let S˜(R+) be the linear space of all measurable, a.e. finite functions f on R+. For any f ∈ S˜(R+) we define its distribution function by d(v; f)=λ(t ∈ R+ : |f(t)| >v)(v ≥ 0), where λ denotes Lebesgue measure. Let S(R+) be the subspace of all f ∈ S˜(R+) such that d(v; f) < ∞ for some v>0 and let S0(R+) be the subspace of all f ∈ S(R+) with d(v; f) < ∞ for all v>0. For f ∈ S(R+)wedenotebyμ(f)the decreasing rearrangement of f, defined by μt(f) = inf{v>0:d(v; f) ≤ t} (t ≥ 0). For f,g ∈ S(R+)wesayf is submajorized by g, and write f ≺≺ g,if t t μs(f)ds ≤ μs(g)ds, for all t>0. 0 0 A (quasi-)normed linear subspace E of S(R+)iscalleda(quasi-)Banach function space on R+ if it is complete and if for f ∈ S(R+)andg ∈ E with |f|≤|g| we have f ∈ E and fE ≤gE. A (quasi-)Banach function space E on R+ is called symmetric if for f ∈ S(R+)andg ∈ E with μ(f) ≤ μ(g)wehavef ∈ E and fE ≤gE . It is called fully symmetric if, in addition, for f ∈ S(R+)andg ∈ E with f ≺≺ g it follows that f ∈ E and fE ≤gE . A symmetric (quasi-)Banach function space is said to have a Fatou (quasi-)norm if for every net (fβ)inE and f ∈ E satisfying 0 ≤ fβ ↑ f we have fβE ↑fE. The space E is said to have the Fatou property if for every net (fβ)inE satisfying ≤ ↑ ∞ ↑ 0 fβ and supβ fβ E < the supremum f =supβ fβ exists in E and fβ E fE.WesaythatE has order continuous (quasi-)norm if for every net (fβ)inE such that fβ ↓ 0wehavefβE ↓ 0. In the literature, a symmetric (quasi-)Banach License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4082 SJOERD DIRKSEN function space is often called rearrangement invariant if it has order continuous (quasi-)norm or the Fatou property. We shall not use this terminology. Let us finally discuss some results specific for symmetric Banach function spaces. The K¨othe dual of a symmetric Banach function space E is the Banach function space E× given by ∞ × E = g ∈ S(R+):sup |f(t)g(t)| dt : fE ≤ 1 < ∞ ; 0 ∞ × gE× =sup |f(t)g(t)| dt : fE ≤ 1 ,g∈ E . 0 The space E× is fully symmetric and has the Fatou property. It is isometrically isomorphic to a closed subspace of E∗ via the map ∞ g → Lg,Lg(f)= f(t)g(t) dt (f ∈ E). 0 A symmetric Banach function space on R+ has a Fatou norm if and only if E embeds isometrically into its second K¨othe dual E×× =(E×)×. It has the Fatou property if and only if E = E×× isometrically. It has order continuous norm if and only if it is separable, which is also equivalent to the statement E∗ = E×.Moreover,a symmetric Banach function space which is separable or has the Fatou property is automatically fully symmetric. For proofs of these facts and more details we refer to [4, 23, 24]. 2.1. Boyd indices. We now discuss the Boyd indices, which were introduced by D.W. Boyd in [5]. For any 0 (Daf)(s)=f(as)(s ∈ R+). The following lemma is well known for symmetric Banach function spaces (cf. [23]). Lemma 2.2. Let E be a symmetric quasi-Banach function space on R+.Then,for every 0 Dbμ(f)(s)=μbs(f) ≤ μas(f)=Daμ(f)(s). Hence, if Da is bounded on E,thenDb is bounded on E as well and Db≤Da. In particular, Da is bounded on E if a ≥ 1andDa≤1. Moreover, it suffices to show that D 1 is bounded on E for every n ∈ N. ∈ N n ∈ ≤ ≤ Fix n ,letf E+ and let fi,1 i n, be mutually disjoint functions n having the same distribution function as f.ThenD 1 f and fi have the same n i=1 distribution function. Indeed, λ(t ∈ R+ :(D 1 f)(t) >v)=nλ(t ∈ R+ : f(t) >v) n n = λ(t ∈ R+ : fi(t) >v) i=1 n (2) = λ t ∈ R+ : fi(t) >v . i=1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NONCOMMUTATIVE BOYD INTERPOLATION THEOREMS 4083 Since E is symmetric it follows that D 1 f ∈ E. Moreover, by Theorem 2.1, there n exists some c>0and0 n n 1 p p 1 ≤ p (3) D 1 f E = fi c fi E = cn f E . n E i=1 i=1 From the above it is clear that a →Da is decreasing and, since Dab = DaDb if a ≤ b, submultiplicative. Define the lower Boyd index pE of E by log s pE = lim s→∞ log D 1 s and the upper Boyd index qE of E by log s qE = lim . s↓0 log D 1 s By the Aoki-Rolewicz theorem E admits an equivalent p-norm for some 0 1 D 1 fE ≤ n p fE n and therefore p ≤ pE. Inparticularwehave0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4084 SJOERD DIRKSEN or max fiE ≤ C max |fi| (if q = ∞). 1≤i≤n 1≤i≤n E The least constant M(q) for which this inequality holds is called the q-concavity constant of E. It is clear that every quasi-Banach function space is ∞-concave (1) with M(∞) = 1 and any Banach function space is 1-convex with M =1. For 1 ≤ r<∞,letther-concavification and r-convexification of E be defined by 1 1 { ∈ R | | r ∈ } | | r r E(r) := g S( +): g E , g E(r) = g E, 1 (r) { ∈ R | |r ∈ } | |r r E := g S( +): g E , g E(r) = g E, respectively. As is shown in [24] (p. 53), if E is a Banach function space, then E(r) is a Banach function space. In general, E(r) is only a quasi-Banach function space. s s (r) Using that μ(|f| )=μ(f) for any f ∈ S(R+)and0 Example 2.1 (Lorentz spaces Lp,q). Let 0 ∞ q −1 1 p q q ∞ ( 0 t μt(f) dt) (0 Example 2.2 (Orlicz spaces). Let Φ : [0, ∞) → [0, ∞] be a Young’s function, i.e., a convex, continuous and increasing function satisfying Φ(0) = 0 and limt→∞ Φ(t)= ∞.TheOrlicz space LΦ is the subspace of all f in S(R+) such that for some k>0, ∞ |f(t)| Φ dt < ∞. 0 k If we equip LΦ with the Luxemburg norm ∞ | | f(t) ≤ f LΦ =inf k>0: Φ dt 1 , 0 k License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NONCOMMUTATIVE BOYD INTERPOLATION THEOREMS 4085 then LΦ is a symmetric Banach function space with the Fatou property [4,24]. The Boyd indices of LΦ can be computed in terms of Φ. Indeed, let Φ(ts) MΦ(t)=sup , s>0 Φ(s) and define the Matuszewska-Orlicz indices by log MΦ(t) log MΦ(t) pΦ = lim ,qΦ = lim . t↓0 log t t→∞ log t One can show that pΦ = pLΦ and qΦ = qLΦ ; see e.g. the proof of [27], Theorem 4.2. For our discussion of Φ-moment inequalities we will need the following results on Orlicz functions. We say that an Orlicz function satisfies the global Δ2-condition if for some constant C>0, (6) Φ(2t) ≤ CΦ(t)(t ≥ 0). Under this condition we have, for any α ≥ 0, Φ(αt) α,Φ Φ(t)(t ≥ 0). One can show ([27], Theorem 3.2(b)) that (6) is equivalent to the assumption qΦ < ∞, which in turn holds if and only if tΦ(t) (7) sup < ∞. t>0 Φ(t) Finally, we shall use the following characterization of Boyd’s indices for Orlicz spaces ([27], Theorem 6.4): t −p ds −p pΦ =sup p>0: s Φ(s) p,Φ t Φ(t) ∀t>0 , 0 s (8) ∞ −q ds −q qΦ =inf q>0: s Φ(s) q,Φ t Φ(t) ∀t>0 . t s We refer to [4, 23, 24] for many more concrete examples of symmetric quasi- Banach function spaces. 3. Characterization of Marcinkiewicz weak type operators In this section we establish a key observation which essentially reduces the proof of Boyd’s theorem and its noncommutative extensions to proving a certain inequal- ity for distribution functions, which is stated in Lemma 3.7 below. This observation moreover leads to a characterization of the subconvex operators which are simulta- neouslyofweaktypes(p, p)and(q, q); see Theorem 3.8. For 0 θp,q(t)=ψp(t)+φq(t)(t>0). Here it is understood that φ∞ = χ(0,1). Corresponding to these functions we define three linear operators Φq, Ψp, Θp,q : S(R+) → S˜(R+ × R+)by Φq(f)=f ⊗ φq, Ψp(f)=f ⊗ ψp, Θp,q(f)=f ⊗ θp,q. The following observation is a reformulation of [7], Lemma 4.3. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4086 SJOERD DIRKSEN Lemma 3.1. Let E be a symmetric quasi-Banach function space on R+ and let 0 Clearly Φ∞ is an isometry from E(R+)intoE(R+ × R+) for every symmetric quasi-Banach function space E. The corresponding result for the lower Boyd index reads as follows. In the proof and later on, we use χA to denote the indicator of a set A. Lemma 3.2. Let E be a symmetric quasi-Banach function space on R+ and let 0 Proof. Fix p for any f ∈ E+.Observethatf ⊗ χ[2n,2n+1) has the same distribution on R+ × R+ as D2−n f on R+. Hence, ∞ − n ⊗ ≤ p n n+1 f ψp E(R+×R+) f(s) 2 χ[2 ,2 )(t) E(R ×R ) n=0 + + ∞ 1 − nr r r ≤ C 2 p f(s)χ n n+1 (t) [2 ,2 ) E(R+×R+) n=0 ∞ 1 − nr r r = C 2 p D −n f , 2 E(R+) n=0 ≤ where C and 0 ∞ 1 − nr nr r r f ⊗ ψ R ×R ≤ cC 2 p 2 p0 f p E( + +) p0 E(R+) n=0 p,E f E(R+), as 1 − 1 < 0. p0 p For the second assertion, notice first that μ(Ds(f)) = Dsμ(f) for all 0 where we use that f ⊗ χ[a,2a) has the same distribution on R+ × R+ as Da−1 f on R+. By (9) we arrive at 1 1 1 −1 ≤ p p ⊗ p Da f E 2 a f ψp E(R+×R+) p,E a f E. Since this holds for any 1 ≤ a<∞, we conclude that p ≤ pE. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NONCOMMUTATIVE BOYD INTERPOLATION THEOREMS 4087 As a result of Lemmas 3.1 and 3.2 we find the following novel expressions for Boyd’s indices: pE =sup p>0:∃C>0 ∀f ∈ E Ψp(f)E(R ×R ) ≤ CfE(R ) , + + + ∃ ∀ ∈ ≤ qE =inf q>0: C>0 f E Φq(f) E(R+×R+) C f E(R+) . Moreover, we have the following result. Corollary 3.3. Let 0 The bound for the operator norm of Θp,q given in the proof of Corollary 3.3 can be improved in specific situations. For example, if 0 1 p q r Θ r → r = + . p,q L L r − p q − r We now compute the distribution function of Φq(f), Ψp(f)andΘp,q(f). The first was already done in [7], Lemma 4.4. Lemma 3.4. Let 0 and d(v;Φ∞(f)) = d(v; f). Lemma 3.5. Let 0 Proof. Using a change of variable, λ (s, t) ∈ R+ × R+ : f(s)ψp(t) >v ∞ − 1 = λ s ∈ R+ : f(s)t p >v dt 1 ∞ f(s) 1 = λ s ∈ R+ : >tp dt 1 v ∞ f(s) p−1 = λ s ∈ R+ : >u pu du 1 v p 1 f f(s) p−1 = − λ s ∈ R+ : >u pu du. p R v L ( +) 0 v License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4088 SJOERD DIRKSEN Observe that 1 f(s) p−1 λ s ∈ R+ : >u pu du 0 v ∞ f(s) = λ s ∈ R :min , 1 >u pup−1du + v 0 p −1 p f(s) = min{v f,1} p = ds + d(v; f), L (R+) {f≤v} v which gives the conclusion. Corollary 3.6. Let 0 Proof. Since f ⊗ φq and f ⊗ ψp have disjoint supports we have d(v;Φq(f)) + d(v;Ψp(f)) = d(v;Θp,q(f)). Therefore, if d(v; f) < ∞, then (10) and (11) follow immediately from Lemmas 3.4 and 3.5. On the other hand, for any v>0, d(v; f) ≤ d(v;Φq(f)) ≤ d(v;Θp,q(f)) and f(s) p d(v; f) ≤ ds. {f>v} v Hence, if d(v; f)=∞, then both sides of (10) and (11) are equal to ∞. Lemma 3.7. Let E be a symmetric quasi-Banach function space on R+.Letα>0 and f ∈ E+. Suppose that either p (12) d(αv; g) ≤ d(v;Θp,qf)(v>0). Then g ∈ E and gE ≤ αΘp,qE→E fE. Proof. We take right continuous inverses in (12) to obtain μt(g) ≤ αμt(Θp,qf)(t ≥ 0). As E is symmetric, it follows from Corollary 3.3 that g ∈ E and, moreover, ≤ ≤ g E α Θp,qf E(R+×R+) α Θp,q E→E f E. The following result is reminiscent of Calder´on’s characterization of weak type operators (see Theorem A.1 for a noncommutative extension). Recall that if D is aconvexsetinS(R+), then an operator T : D → S(R+) is called subconvex if for any f,g ∈ D and t ∈ [0, 1] we have T (tf +(1− t)g) ≤ tT (f)+(1− t)T (g). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NONCOMMUTATIVE BOYD INTERPOLATION THEOREMS 4089 p Theorem 3.8. Let 0 if and only if there is some α>0 such that for all f ∈ S(R+), (14) d(αv; Tf) ≤ d(v;Θp,q(f)) (v>0). Proof. Suppose that (13) holds and fix v>0. We may assume that d(v;Θp,q(f)) < ∞, for otherwise there is nothing to prove. By Corollary 3.6 it follows that p q fχ{f>v} ∈ L (R+)andfχ{f≤v} ∈ L (R+). If Cp,q =max{Cp,Cq},thenby subconvexity, ≤ 1 1 d(2Cp,qv; Tf) d(2Cp,qv; 2 T (2fχ{f≤v})+ 2 T (2fχ{f>v})) ≤ d(2Cp,qv; T (2fχ{f≤v})) + d(2Cp,qv; T (2fχ{f>v})). By (13) and Corollary 3.6, d(2Cp,qv; Tf) ≤ −q q q −p p p (2Cp,qv) C 2fχ{ ≤ } q +(2Cp,qv) C 2fχ{ } p q f v L (R+) p f>v L (R+) ≤ d(v;Θp,qf). Suppose now that (14) holds. If q<∞, then by Corollary 3.6, −q q −p p d(αv; Tf) ≤ d(v;Θp,q(f)) = v f(s) ds + v f(s) ds. {f≤v} {f>v} Since p ≤ q we have −1 p −1 q −1 q −1 p (v f) χ{f>v} ≤ (v f) χ{f>v}, (v f) χ{f≤v} ≤ (v f) χ{f≤v}, and therefore −r r d(αv; Tf) ≤ v f r (r = p, q). L (R+) On the other hand, if q = ∞,thenitisclearthat ≤ −p p ≤ −p p d(αv; Tf) d(v;Θp,∞(f)) = v f(s) ds v f p . L (R+) {f>v} Moreover, for any v>0wehave d(αv; T (fχ{f≤v})) = 0. Applying this for v = f∞ yields Tf ≤ αf∞ a.e. This completes the proof. The following result shows that inequality (12) also implies Φ-moment inequali- ties. Lemma 3.9. Let Φ be an Orlicz function on R+ which satisfies the global Δ2- condition. Let α>0 and f ∈ (LΦ)+. Suppose that either p (15) Φ(|g(t)|)dt p,q,Φ Φ(f(t))dt. 0 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4090 SJOERD DIRKSEN Proof. Suppose that qΦ Φ(|g(t)|)dt p,q,Φ Φ(t)dλf (t)= Φ(f(t))dt. 0 0 0 The statement for q = ∞ is proved analogously. Remark 3.10. From the presented proof it is clear that the result in Lemma 3.9, and hence the Φ-moment inequalities discussed below, remain valid if Φ is nonconvex, provided that it satisfies (6) and (7), and pΦ,qΦ are understood as in (8). It should be noted that in this case LΦ is in general no longer a quasi-Banach space. Boyd’s interpolation theorem for Marcinkiewicz weak type operators, as well as a Φ-moment version, are now an immediate consequence of the previous observations. p q Theorem 3.11. Fix 0 On the other hand, if Φ is an Orlicz function on R+ satisfying the global Δ2- condition and either p d(2 max{Cp,Cq}v; Tf) ≤ d(v;Θp,qf). The assertions now follow from Lemmas 3.7 and 3.9, respectively. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NONCOMMUTATIVE BOYD INTERPOLATION THEOREMS 4091 Remark 3.12. Theorem 3.11 can be extended to a vector-valued interpolation the- orem using the following simple trick contained in the proof of [3], Lemma 1. Sup- p,1 q,1 pose that X, Y are Banach spaces and T : L (R+; X)+L (R+; X) → S(R+; Y ) satisfies r,1 r,∞ ≤ r,1 ∈ R Tf L (R+;Y ) Cr f L (R+;X) (f L ( +; X),r= p, q). For a fixed f set k(f)=(f/fX )χ{f=0 } and define the subconvex operator p,1 q,1 Sg = T (gk(f))Y (g ∈ L (R+)+L (R+)). Since k(f)X = 1, it follows that r,1 r,∞ ≤ r,1 r,1 ∈ R Sg L (R+) Cr gk(f) L (R+;X) = Cr g L (R+) (g L ( +),r= p, q), and hence by the scalar-valued Boyd interpolation theorem, SgE ≤ 2max{Cp,Cq}Θp,qgE (g ∈ E). Taking g = fX yields ≤ { } ∈ R Tf E(R+;Y ) 2max Cp,Cq Θp,q f E(R+;X) (f E( +; X)). 4. Noncommutative Boyd interpolation theorems In this section we prove a noncommutative version of Boyd’s theorem, Theo- rem 4.8 below. We first recall some terminology and preliminary results for non- commutative symmetric spaces. Let M be a semi-finite von Neumann algebra acting on a complex Hilbert space H, which is equipped with a normal, semi-finite, faithful trace τ.Thedistribution function of a closed, densely defined operator x on H, which is affiliated with M,isgivenby d(v; x)=τ(e|x|(v, ∞)) (v ≥ 0), where e|x| is the spectral measure of |x|.Thedecreasing rearrangement of x is defined by μt(x) = inf{v>0:d(v; x) ≤ t} (t ≥ 0). We say that x is τ-measurable if d(v; x) < ∞ for some v>0. We let S(τ)be the linear space of all τ-measurable operators, which is a metrizable, complete topological ∗-algebra with respect to the measure topology. We denote by S0(τ) the linear subspace of all x ∈ S(τ) such that d(v; x) < ∞ for all v>0. One can introduce a partial order on the linear subspace S(τ)h of all self-adjoint operators in S(τ) by setting, for a self-adjoint operator x, x ≥ 0 if and only if xξ, ξH ≥ 0 for all ξ ∈ D(x), where D(x) is the domain of x in H.Wewritex ≤ y for x, y ∈ S(τ)h if and only if y − x ≥ 0. Under this partial ordering S(τ)h is a partially ordered vector space. Let S(τ)+ denote the positive cone of all x ∈ S(τ)h satisfying x ≥ 0. It can be shown that S(τ)+ is closed with respect to the measure topology ([11], Proposition 1.4). Throughout our exposition, we will tacitly use many properties of distribution functions and decreasing rearrangements. For the convenience of the reader we collect these facts in the following two propositions. The first result is essentially contained in the proof of [30], Theorem 1. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4092 SJOERD DIRKSEN Proposition 4.1. If x, y ∈ S(τ),then: (a) d(v; x)=d(v; μ(x)) for all v ≥ 0; (b) d(v + w; x + y) ≤ d(v; x)+d(w; y) for all v, w ≥ 0; (c) if |x|≤|y|,thend(v; x) ≤ d(v; y) for all v ≥ 0. The following properties of decreasing rearrangements can be found in [13]. If p is a projection in M,thenweletp⊥ := 1 − p denote its orthogonal complement. Proposition 4.2. If x, y ∈ S(τ),then: (a) μt(λx)=|λ|μt(x) for all λ ∈ C and t ≥ 0; (b) μs+t(x + y) ≤ μs(x)+μt(y) for all s, t ≥ 0; (c) if |x|≤|y|,thenμt(x) ≤ μt(y) for all t ≥ 0; (d) μt(uxv) ≤u μt(x) y for all u, v ∈Mand t ≥ 0. If e = e|x|(v, ∞),then (e) μt(|x|e)=μt(x)χ[0,τ(e))(t) for all t ≥ 0; ⊥ (f) μt(|x|e )=μt+τ(e)(x) for all t ≥ 0,providedτ(e) < ∞. Finally, suppose that φ :[0, ∞) → [0, ∞) is an increasing function which is left- continuous on (0, ∞) and satisfies φ(0) = 0. If we define φ(∞) := limt→∞ φ(t), then (g) μ(φ(|x|)) = φ(μ(x)) on [0, ∞). For a symmetric (quasi-)Banach function space E on R+, we define E(M,τ):={x ∈ S(τ): μ(x)E < ∞}. We usually denote E(M,τ)byE(M) for brevity. We call E(M)thenoncommuta- tive (quasi-)Banach function space associated with E and M. In the quasi-Banach case these spaces were first considered by Xu in [35]. The following fundamental result is proved in [22], Theorem 8.11 (see also [11, 35] for earlier proofs of this result under additional assumptions). Theorem 4.3. If E is a symmetric (quasi-)Banach function space on R+ which is p-convex for some 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NONCOMMUTATIVE BOYD INTERPOLATION THEOREMS 4093 −2/s M ∈ M it follows that n≥1 n xn converges in E( )tosomex E( )+, and since p q p q −2/s E(M) ⊂ L (M)+L (M)wehavex ∈ (L (M)+L (M))+. But n xn ≤ x and −2/s so n Lemma 4.5. Let 0 Proof. If x ∈ E(M), then by Corollary 3.3 we have Θp,qμ(x) ∈ E and hence |x| d(v;Θp,qμ(x)) < ∞ for some v>0. If ev = e [0,v], then by Proposition 4.2 q q ⊥p p xev Lq (M) = μt(x) dt, xev Lp(M) = μt(x) dt. {μ(x)≤v} {μ(x)>v} It therefore follows from Corollary 3.6 that −q q −p ⊥p ∞ v xev Lq (M) + v xev Lp(M) = d(v;Θp,qμ(x)) < . Hence x ∈ Lp(M)+Lq(M). By Lemma 4.4 this implies that E(M) ⊂ Lp(M)+ Lq(M) continuously. Suppose now that q = ∞. Pick v>0 such that d(v;Θp,∞μ(x)) < ∞.Then ⊥ p xev ∈Mand xe ∈ L (M) since by Proposition 4.2 and Corollary 3.6, v −p ⊥p −p p v xev Lp(M) = v μt(x) dt = d(v;Θp,∞μ(x)). {μ(x)>v} By Lemma 4.4 we conclude that E(M) embeds continuously into Lp(M)+M. The first inclusion is immediate from the commutative case (see e.g. [24], Propo- sition 2.b.3), as (Lp ∩ Lq)(M)=Lp(M) ∩ Lq(M). To formulate our main result the following definition is convenient. The notion of subconvexity given below weakens the notion of sublinear operators on spaces of measurable operators introduced by Q. Xu (see [14], where it first appeared in published form). Definition 4.6. Let M and N be von Neumann algebras equipped with normal, semi-finite, faithful traces τ and σ, respectively. Let D be a convex subset of S(τ). AmapT : D → S(σ)h is called midpoint convex if 1 1 ≤ 1 1 T ( 2 x + 2 y) 2 T (x)+ 2 T (y) for all x, y ∈ D.AmapU : D → S(σ) is called midpoint subconvex if for every x, y ∈ D there exist partial isometries u, v ∈N such that | 1 1 |≤ 1 ∗| | 1 ∗| | U( 2 x + 2 y) 2 u Ux u + 2 v Uy v. It is a well-known fact (see e.g. [13], Lemma 4.3) that for any x, y ∈ S(σ)thereare partial isometries u, v ∈N such that |x + y|≤u∗|x|u + v∗|y|v. Therefore, any linear map is (midpoint) subconvex. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4094 SJOERD DIRKSEN For further reference we state Chebyshev’s inequality. Lemma 4.7 (Chebyshev’s inequality). Let 0 Theorem 4.8. Let E be a symmetric quasi-Banach function space on R+ which is s-convex for some 0 If p TxE(N ) ≤ 2Θp,q max{Cp,Cq}xE(M) (x ∈ E(M)+). p q The same result holds if T : L (M)+ + L (M)+ → S(σ)h is a midpoint convex map satisfying (17). Remark 4.9. Property (17) is clearly the noncommutative version of Marcinkiewicz weak type (r, r). The reader should be warned, however, that in the noncommuta- tive literature it is nowadays customary to simply refer to this property as ‘weak type (r, r)’. Proof. We may assume that max{Cp,Cq}≤1. By Lemma 4.5 T is well defined x on E(M)+.Letx ∈ E(M)+ and let ev = e [0,v]. By midpoint subconvex- ∈N | |≤ 1 ∗| | ity, there exist partial isometries u1,u2 such that Tx 2 u1 T (2xev) u1 + 1 ∗| ⊥ | 2 u2 T (2xev ) u2. It follows that ≤ 1 ∗| | 1 ∗| ⊥ | d(2v; Tx) d(v; 2 u1 T (2xev) u1)+d(v; 2 u2 T (2xev ) u2) ≤ ⊥ (18) d(2v; T (2xev)) + d(2v; T (2xev )). Suppose first that qE = d(v;Θp,qμ(x)). The result now follows from Lemma 3.7, using that d(v; Tx)=d(v; μ(Tx)). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NONCOMMUTATIVE BOYD INTERPOLATION THEOREMS 4095 Suppose now that q = ∞.Then ∗ 1 ∞ ≤ ∞ ≤ 2 u1T (2xev)u1 L (N ) C∞ xev L (M) v, 1 ∗ so d(v; 2 u1T (xev)u1) = 0. By (16) and (17) we have ≤ −p p p ∈ p M d(v; Ty) v Cp y Lp(M) (v>0,y L ( )+), and therefore (18) implies that d(2v; Tx) ≤ Cp(2v)−p2xe⊥p p v Lp(M) −p p ≤ v μt(x) dt = d(v;Θp,∞μ(x)). {μ(x)>v} Lemma 3.7 gives the conclusion. It is clear from the proof of Theorem 4.8 that the same result holds for mid- p q p q point (sub)convex operators on L (M)h + L (M)h or L (M)+L (M) instead of p q L (M)+ + L (M)+. The original version of Boyd’s theorem allows for the interpolation of operators of weak-type (p, p), i.e., which are bounded from Lp,1 into Lp,∞. Theorem 4.8 only applies for Marcinkiewicz weak type (p, p) operators. In Theorem A.3 in the appendix we will show how to obtain a full noncommutative analogue of Boyd’s theorem using a different approach. 4.1. Interpolation of noncommutative probabilistic inequalities. To illus- trate the flexibility of the method used to prove Theorem 4.8, we modify it to inter- polate several noncommutative probabilistic inequalities. In particular we prove the dual version of Doob’s maximal inequality in noncommutative symmetric spaces; see Corollary 4.13 below. The latter result is a consequence of Theorem 4.11, which we will interpret in the following section as an interpolation result for operators on noncommutative 1-valued symmetric spaces. For its proof, we shall need the fol- lowing observation. Lemma 4.10. Let x ∈ S(τ)+.Ife is a projection in M,then x ≤ 2(exe + e⊥xe⊥). Proof. By writing x = exe + e⊥xe + exe⊥ + e⊥xe⊥, we see that the asserted inequality is equivalent to exe − e⊥xe − exe⊥ + e⊥xe⊥ ≥ 0. But x ≥ 0, so ⊥ ⊥ ⊥ ⊥ 1 1 ⊥ ∗ 1 1 ⊥ exe − e xe − exe + e xe =(x 2 e − x 2 e ) (x 2 e − x 2 e ) ≥ 0, and the result follows. Theorem 4.11. Let E be a symmetric quasi-Banach function space on R+ which is s-convex for some 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4096 SJOERD DIRKSEN convex maps such that for some constants Cp,Cq > 0 depending only on p and q, respectively, r (19) Tk(xk) ≤ Cr xk (xk ∈ L (M)+,k ≥ 1,r= p, q). Lr,∞(N ) Lr (M) k≥1 k≥1 If p = d(v;Θp,qμ(x)), where the final equality follows from Corollary 3.6. The result is now immediate from Lemma 3.7. The case q = ∞ follows analogously as in the proof of Theo- rem 4.8. As an application of Theorem 4.11, we can interpolate the following dual Doob inequality in noncommutative Lp-spaces, due to M. Junge. Theorem 4.12 ([17]). Let M be a semi-finite von Neumann algebra and let (Ek)k≥1 be an increasing sequence of conditional expectations in M.If1 ≤ p<∞, then for p any sequence (xk)k≥1 in L (M)+, Ek(xk) p xk . Lp(M) Lp(M) k k Theorems 4.12 and 4.11 together yield the following extension. Corollary 4.13. Let E be a symmetric quasi-Banach function space on R+ which is s-convex for some 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NONCOMMUTATIVE BOYD INTERPOLATION THEOREMS 4097 Let (Ek)k≥1 be an increasing sequence of conditional expectations in M.If1 < p ≤ q < ∞, then for any sequence (x ) ≥ in E(M) , E E k k 1 + (21) Ek(xk) E xk , E(M) E(M) k≥1 k≥1 where the sums converge in norm. In [18], Theorem 7.1, it was shown that any conditional expectation E is ‘anti- bounded’ for the Lp-norm if 0 xE(M) E E(x)E(M) (x ∈M). y Proof. Let y = E(x)andforv>0setev = e [0,v]. As was remarked after Lemma 2.2, we have pE > 0 and hence we can fix 0 = d(v;Θp,qμ(y)), where the final equality follows from Corollary 3.6. The conclusion now follows from Lemma 3.7. The following result facilitates interpolation of noncommutative square function estimates. Theorem 4.15. Let E be a symmetric quasi-Banach function space on R+ which is s-convex for some 0 If p License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4098 SJOERD DIRKSEN 1 ≤ | |2 2 Proof. We may assume Cp,Cq 1. Let x =( k≥1 xk ) and for v>0 define x ev = e [0,v]. Then, ≤ ⊥ d 2v; Tk(xk) d v; Tk(xkev ) + d v; Tk(xkev) . k≥1 k≥1 k≥1 By (23), 1 1 2 p 2 q ≤ −p | ⊥|2 −q | |2 d 2v; Tk(xk) v xkev + v xkev Lp(M) Lq (M) k≥1 k≥1 k≥1 1 1 2 p 2 q −p | |2 ⊥ −q | |2 = v xk ev + v xk ev Lp(M) Lq (M) k≥1 k≥1 −p p −q q = v μt(x) dt + v μt(x) dt {μ(x)>v} {μ(x)≤v} = d(v;Θp,qμ(x)). The result now follows from Lemma 3.7. The case q = ∞ is similar. As a corollary, we find the following version of Stein’s inequality for noncommu- tative symmetric spaces, which will be needed in the proof of Theorem 6.2 below. A different proof of this result was found in [16]. Corollary 4.16. Let E be a symmetric quasi-Banach function space on R+ which is s-convex for some 0 1 1 2 2 2 2 (24) |Ek(xk)| E |xk| . E(M) E(M) k≥1 k≥1 Proof. Let ekl be the standard matrix units, let yk = xk ⊗ ek1 and let Tk = Ek ⊗ 1B(2). Then (24) is equivalent to 1 2 2 Tk(yk) E |yk| . E(M⊗B(2)) E(M⊗B(2)) k≥1 k≥1 By [32], Theorem 2.3, this inequality holds if E = Lp with 1 Further examples of probabilistic inequalities which can be interpolated using the presented method are given by the ‘upper’ noncommutative Khintchine inequalities; see [7], Theorem 4.1, and [8], Corollary 2.2. Remark 4.17. The results in Theorems 4.8, 4.11 and 4.15, Corollaries 4.13 and 4.16, and Proposition 4.14 all have an appropriate ‘Φ-moment version’. Indeed, these versions follow immediately by using Lemma 3.9 instead of Lemma 3.7 in the proofs of the latter results. In particular, by following the proof of Theorem 4.8 and taking Remark 3.10 into account, we find an extension of [1], Theorem 2.1, for nonconvex Orlicz functions. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NONCOMMUTATIVE BOYD INTERPOLATION THEOREMS 4099 5. Interpolation of noncommutative maximal inequalities In this section we present a Boyd-type interpolation theorem for noncommutative maximal inequalities. To formulate maximal inequalities in noncommutative sym- metric spaces and their dual versions, we first introduce the two ‘noncommutative vector-valued symmetric spaces’ E(M; ∞)andE(M; 1). We define these in anal- ogy with the noncommutative vector-valued Lp-spaces Lp(M; ∞)andLp(M; 1), which were introduced in [31] for hyperfinite von Neumann algebras and considered in general in [17]. From now on, we let E be a symmetric Banach function space on R+. ∞ We define E(M; ) to be the space of all sequences x =(xk)k≥1 in E(M)for (2) which there exist a, b ∈ E (M) and a bounded sequence y =(yk)k≥1 such that xk = aykb (k ≥ 1). For x ∈ E(M; ∞) we define ∞ { } (25) x E(M; ) =inf a E(2)(M) sup yk ∞ b E(2)(M) , k≥1 where the infimum is taken over all possible factorizations of x as above. We can think of the quantity (25) as ‘ supk≥1 xk E(M)’, even though supk≥1 xk need not be defined at all. 1 We define E(M; ) to be the space of all sequences x =(xk)k≥1 in E(M)which can be decomposed as ∗ ≥ xk = ujkvjk (k 1) j≥1 (2) for two families (ujk)j,k≥1 and (vjk)j,k≥1 in E (M) satisfying ∗ ∈ M ∗ ∈ M ujkujk E( )and vjkvjk E( ), j,k j,k where the series converge in norm. For x ∈ E(M; 1) we define 1 1 2 2 ∗ ∗ x E(M;1) =inf ujkujk vjkvjk , E(M) E(M) j,k j,k where the infimum is taken over all decompositions of x as above. In what follows, 1 we will mostly consider elements x =(xk)k≥1 ∈ E(M; ) for which xk ≥ 0 for all k.Inthiscase, xE(M;1) = xk . E(M) k≥1 The theory for the spaces E(M; ∞)andE(M; 1) can be developed in full analogy with the special case E = Lp considered in [17, 19, 34]. In fact, most of the basic p p × 1 1 results follow verbatim as soon as we replace L by E, L by E ,where p + p = 1, and L2p by E(2) in the proofs of these results. For example, the following observation is immediate. ∞ Theorem 5.1. If E is a symmetric Banach function space on R+,thenE(M; ) and E(M; 1) are Banach spaces. Our strategy to prove a Boyd-type interpolation theorem for maximal inequalities is to dualize Theorem 4.11, which can be viewed as a Boyd-type interpolation theorem for 1-valued noncommutative symmetric spaces. We shall need the duality License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4100 SJOERD DIRKSEN stated in Theorem 5.3 below. The proof is essentially an adaptation of the Hahn- Banach separation argument in [17], Proposition 3.6 (see also [34], Theorem 4.11) to our context. We need the following observation, proved in [11], Theorem 5.6 and p. 745. Theorem 5.2. If E is a separable symmetric Banach function space on R+,then E(M)∗ = E×(M) isometrically, with associated duality bracket given by x, y = τ(xy)(x ∈ E(M),y∈ E×(M)). Below we will implicitly use the trace property a number of times; i.e., we will use that if x, y ∈ S(τ)aresuchthatxy, yx ∈ L1(M), then τ(xy)=τ(yx). In particular this holds if x ∈ E(M)andy ∈ E×(M). Theorem 5.3. Let M be a semi-finite von Neumann algebra and let E be a sepa- × ∞ rable symmetric Banach function space on R+.Ify =(yk) ∈ E (M; ) satisfies y ≥ 0 for all k,then k (26) yE×(M;∞) =sup τ(xkyk):xk ∈ E(M)+, xk ≤ 1 . E(M) k≥1 k≥1 Proof. We let S denote the supremum on the right hand side of (26). Let yk = azkb × (2) with a, b ∈ (E ) (M)andzk ∈Mwith zk∞ ≤ 1andlet(xk) be a sequence in E(M)+.ByH¨older’s inequality, 1 1 2 2 τ(xkyk)= τ(xkazkb)= τ(bxk xk azk) k k k 1 1 ≤ 2 2 bxk L2(M) xk azk L2(M) k 1 1 ∗ 2 ∗ 2 ≤ τ(bxkb ) τ(a xka) k k ≤ b (E×)(2)(M) xk a (E×)(2)(M). E(M) k We conclude that S ≤yE×(M;∞). Suppose now that S = 1; we will show that yE×(M;∞) ≤ 1. Under this assumption, we have for any finite sequence x =(x )inE(M) , k + (27) τ(xkyk) ≤ xk . E(M) k k × ∗ Let K = {s ∈ E (M)+ : sE×(M) ≤ 1}, equipped with the weak topology of E(M)∗.SinceE×(M) is isometrically isomorphic to E(M)∗ by Theorem 5.2 × ∗ × and E (M)+ is weak closed in E (M), we conclude that K is compact by the Banach-Alaoglu theorem. Moreover, wE(M) =supτ(ws)(w ∈ E(M)+). s∈K For any finite sequence x as above we define fx(s)= τ(xks) − τ(xkyk). k k Clearly fx is a real-valued continuous function on K, and from (27) it follows ≥ that sups∈K fx(s) 0. Let A be the subset of C(K) consisting of all fx,where License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NONCOMMUTATIVE BOYD INTERPOLATION THEOREMS 4101 x =(xk)isafinitesequenceinE(M)+.ThenA is a cone in C(K). Indeed, if λ ≥ 0, then λfx = fλx.Moreover,ifx, x˜ are finite families in E(M), then fx + fx˜ can be realized as fx+˜x, since without loss of generality we may assume thatx ˜ is to the right of the finite family x.ObservethatA is disjoint from the cone A− = {g ∈ C(K):supg<0}. By the Hahn-Banach separation theorem, there exists a real Borel measure μ on K and α ∈ R such that for all f ∈ A and g ∈ A−, gdμ ≤ α ≤ fdμ. K K Note that α =0,asbothA and A− are cones. If B is a Borel subset, then we can find a sequence (gi)inA− such that gn ↑−χB. This shows that μ must be positive, and by normalization we may assume that μ is a probability measure. Hence, for all k ≥ 1, we have (28) τ(xyk) ≤ τ(xs)dμ(s)(x ∈ E(M)+). K Define a positive operator by a = sdμ(s). K × Clearly a ∈ K,soa ∈ E (M)+ and aE×(M) ≤ 1. By (28) and normality of τ, τ(xyk) ≤ τ(xa)(x ∈ E(M)+). This implies that yk ≤ a, and therefore we find a contraction uk ∈Msuch that 1 1 1 ∗ 1 2 2 2 2 yk = uka .Inparticular,yk = a ukuka and hence yE×(M;∞) ≤aE×(M) ≤ 1. This completes the proof. Remark 5.4. Using a slightly more involved version of the separation argument in the proof of Theorem 5.3 (see the proof of [17], Proposition 3.6, for the case E = Lp), one may show that in fact E(M; 1)∗ = E×(M; ∞) isometrically, with respect to the duality bracket x, y = τ(xkyk), k≥1 where x ∈ E(M; 1)andy ∈ E×(M; ∞). The following result facilitates the interpolation of noncommutative maximal inequalities. Theorem 5.5. Let E be the K¨othe dual of a separable symmetric Banach function p,1 q,1 space on R+. Suppose that 1 ≤ p If qE (30) (Sk(x))k≥1E(M;∞) E xE(M) (x ∈ E(M)+). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4102 SJOERD DIRKSEN × Proof. Let F be the symmetric space on R+ such that F = E.By(5)wehave ∗ pF >q,andifpE > 1 also qF ≤ ∞ y Lr (M;1) (Sk(x))k≥1 Lr (M; ) r y Lr (M;1) x Lr,1(M). It follows that ∗ Sk (yk) p yk . Lr ,∞(M) Lr (M) k≥1 k≥1 Therefore, if (y ) ∈ F (M; 1) satisfies y ≥ 0, then by Theorem 4.11, k k ∗ Sk (yk) F yk . F (M) F (M) k≥1 k≥1 Hence, if x ∈ E(M) ,then + ∗ τ(ykSk(x)) = τ(Sk(yk)x) k≥1 k≥1 ≤ ∗ Sk (yk) x E(M) E yk x E(M). F (M) F (M) k≥1 k≥1 By Theorem 5.3 we conclude that (30) holds. Examples of sequences of operators satisfying the conditions of Theorem 5.5 are established in [19]. We give two examples which yield maximal ergodic inequalities in noncommutative symmetric spaces. Let T : M→Mbe a linear map such that (a) T is a contraction on M; (b) T is positive; 1 (c) τ(T (x)) ≤ τ(x) for all x ∈ L (M) ∩M+. In [19], Theorem 4.1, it is shown that for any 1 (Mk(T )(x))k≥1E(M;∞) E xE(M) (x ∈ E(M)+). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NONCOMMUTATIVE BOYD INTERPOLATION THEOREMS 4103 If T moreover satisfies condition (d),then k (T (x))k≥1E(M;∞) E xE(M) (x ∈ E(M)+). To conclude this section, we prove a version of Doob’s maximal inequality for noncommutative symmetric spaces. First recall the following definitions. Let E be a symmetric Banach function space on R+ and let M be a semi-finite von Neumann algebra. Suppose that (Mk)k≥1 is a filtration, i.e., an increasing sequence of von | E Neumann subalgebras such that τ Mk is semi-finite, and let k be the conditional expectation with respect to Mk. A sequence (yk)inE(M) is called a martingale with respect to (Mk)ifEk(yk+1)=yk for all k ≥ 1. We say that (yk)isfinite if there is an n ≥ 1 such that yk = yn for all k ≥ n. A sequence (xk)inE(M)is called a martingale difference sequence if xk = yk − yk−1 for some martingale (yk), with the convention y0 =0andM0 = C1. It was shown by M. Junge in [17] that for every 1 (31) (Ek(y))k≥1Lp(M;∞) p yLp(M). This result implies the following version for noncommutative symmetric spaces. Theorem 5.7. Let M be a semi-finite von Neumann algebra and let E be the K¨othe dual of a separable symmetric Banach function space on R+ with 1 (32) (Ek(y))k≥1E(M;∞) E yE(M). If (yk)k≥1 is a martingale in E(M),then sup ykE(M) ≤(yk)k≥1E(M;∞) E sup ykE(M). k≥1 k≥1 Proof. The first statement follows immediately from Theorem 5.5 and (31). To prove the second statement, let yk = azkb with (zk)k≥1 a bounded sequence in M and a, b ∈ E(2)(M). Since E is the K¨othe dual of a symmetric space, it has the Fatou property and is hence fully symmetric. Therefore, μ(yk) ≺≺ μ(azk)μ(b) implies that ykE(M) ≤μ(azk)μ(b)E ≤ azk E(2)(M) b E(2)(M) ≤ a E(2)(M) zk ∞ b E(2)(M). Taking the infimum over all decompositions as above gives sup ykE(M) ≤(yk)k≥1E(M;∞). k≥1 For the reverse inequality, observe that E(M) ⊂ Lp(M)+Lq(M)forsome1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4104 SJOERD DIRKSEN 6. Burkholder-Davis-Gundy and Burkholder-Rosenthal inequalities As applications of the noncommutative version of Doob’s maximal inequality and its dual version, we derive versions of the Burkholder-Davis-Gundy inequalities and Burkholder-Rosenthal inequalities in noncommutative symmetric spaces. Let E be a symmetric Banach function space on R+. For any finite martingale difference sequence (xk)inE(M)weset 1 1 2 2 | |2 | ∗|2 (xk) HE = xk ; (xk) HE = xk . c E(M) r E(M) k k These expressions define two norms on the linear space of all finite martingale difference sequences in E(M). The following Burkholder-Davis-Gundy inequalities extend the Burkholder-Gundy inequalities established in [7]. Theorem 6.1. Let E be the K¨othe dual of a separable symmetric Banach function space on R+ and suppose that 1 (x ) ≥ E E (y ) ≥ M ∞ (x ) ≥ E ∩ E . k k 1 Hc +Hr E k k 1 E( ; ) E k k 1 Hc Hr Suppose that, moreover, E is separable. If pE > 1 and either qE < 2 or E is 2-concave, then (y ) ≥ M ∞ (x ) E E . k k 1 E( ; ) E k Hc +Hr On the other hand, if either E is 2-convex and qE < ∞ or 2 (33) (y ) ≥ M ∞ (x ) E ∩ E . k k 1 E( ; ) E k Hc Hr Proof. If F is a symmetric Banach function space with F × = E,thenby(5) 1 The following result generalizes the noncommutative Rosenthal inequalities pre- sented in [7], as well as the Burkholder-Rosenthal inequalities for noncommutative Lp-spaces and Lorentz spaces obtained in [18], Theorem 5.1, and [15], Theorem 3.1, respectively. In fact, Theorem 6.2 positively answers an open question posed in [16], Problem 3.5(2). The proof follows the general strategy of the proof of [7], Theorem 6.3. Let Mn(M) denote the von Neumann algebra of n × n matrices with M n M entries in and for any sequence (xk)k=1 in E( )letdiag(xk)andcol(xk)be the matrices with the xk on the diagonal and first column, respectively, and zeroes elsewhere. Theorem 6.2 (Noncommutative Burkholder-Rosenthal inequalities). Let M be a semi-finite von Neumann algebra. Suppose that E is a symmetric Banach function space on R+ satisfying 2 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NONCOMMUTATIVE BOYD INTERPOLATION THEOREMS 4105 be a martingale difference sequence in E(M) with respect to (Mk). Then, for any n ≥ 1, n n 1 2 n E | |2 xk E max diag(xk)k=1 E(Mn(M)), k−1 xk , E(M) E(M) k=1 k=1 n 1 2 E | ∗|2 (34) k−1 xk . E(M) k=1 Proof. We first prove that the maximum on the right hand side is dominated by q M ≤ ∞ k xk E(M). Recall that L ( )hascotypeq if 2 q< , i.e., n 1 n n q q q M ≤ ⊗ diag(xk)k=1 L (Mn( )) = xk Lq (M) rk xk . Lq (L∞⊗M) k=1 k=1 By interpolating this estimate for q =2andq>qE we obtain n n ⊗ diag(xk)k=1 E(Mn(M)) E rk xk . E(L∞⊗M) k=1 Moreover, by [7], Lemma 4.17, rk ⊗ xk E xk . E(L∞⊗M) E(M) k k ≤ ∞ Since 1 1 1 2 2 E ∗ E ∗ k−1(xkxk) = k−1(xkxk) E(M) E (M) k k (2) 1 1 2 2 ∗ ∗ E xkxk = xkxk . E (M) E(M) k (2) k Therefore, by the noncommutative Burkholder-Gundy inequality ([7], Proposition 4.18) we conclude that 1 1 2 2 E ∗ ∗ k−1(xkxk) E xkxk E xk , E(M) E(M) E(M) k k k ∗ and by applying this to the sequence (xk)weget 1 2 E ∗ k−1(xkxk) E xk . E(M) E(M) k k We now prove the reverse inequality in (34). By the noncommutative Burkholder- Gundy inequality, 1 1 2 2 ∗ ∗ (35) xk E max xkxk , xkxk . E(M) E(M) E(M) k k k License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4106 SJOERD DIRKSEN By the quasi-triangle inequality in E(2)(M)wehave 1 ∗ 2 xkxk E(M) k 1 2 ∗ −E ∗ E ∗ (36) E xkxk k−1(xkxk) + k−1(xkxk) . E (M) E (M) k (2) k (2) 2 2 Notice that (|xk| −Ek−1(|xk| ))k≥1 is a martingale difference sequence in E(2)(M). ≤ ∞ Since 1 1 2 ∗ −E ∗ ∗ −E ∗ 2 xkxk k−1(xkxk) E (xkxk k−1(xkxk)) E (M) E (M) k (2) k (2) 1 1 4 2 2 2 2 E |xk| + (Ek−1|xk| ) , E (M) E (M) k (2) k (2) M 2 where in the final inequality we use the quasi-triangle inequality in E(2)( ; c ). By applying the noncommutative Stein inequality (Corollary 4.16) to the second term on the right hand side, we find that 1 2 ∗ −E ∗ | |4 xkxk k−1(xkxk) E xk . E (M) E (M) k (2) k (2) Let x =col(|xk|)andy =diag(|xk|). Since μ(yx) ≺≺ μ(x)μ(y), it follows from the Calder´on-Mitjagin theorem ([23], Theorem II.3.4) that there is a contraction T for thecouple(L1,L∞) such that μ(yx)=T (μ(x)μ(y)). Therefore, 1 4 2 ∗ ∗ 1 | | 2 xk = (x y yx) E (Mn(M)) E (M) (2) k (2) = yx E(2)(Mn(M)) E μ(x)μ(y) E(2) 1 1 2 2 2 ≤ = μ(x) μ(y) E y E(Mn(M)) x E(Mn(M)) 1 2 | |2 (37) = diag(xk) E(Mn(M)) xk , E(M) k where in the final inequality we use H¨older’s inequality. Putting our estimates together, starting from (36), we arrive at 1 2 2 |xk| E(M) k 1 1 1 2 2 2 2 | |2 E | |2 E diag(xk) E(Mn(M)) xk + k−1 xk . E(M) E(M) k k 1 | |2 2 In other words, if we set a = ( k xk ) E(M), b = diag(xk) E(Mn(M)) and 1 E | |2 2 2 2 c = ( k k−1 xk ) E(M),wehavea E ab+c . Solving this quadratic equation we obtain a E max{b, c} or 1 1 2 2 | |2 E | |2 xk E max diag(xk) E(Mn(M)), k−1 xk . E(M) E(M) k k License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NONCOMMUTATIVE BOYD INTERPOLATION THEOREMS 4107 ∗ Applying this to the sequence (xk)gives 1 1 2 2 | ∗|2 E | ∗|2 xk E max diag(xk) E(Mn(M)), k−1 xk . E(M) E(M) k k The result now follows by (35). Remark 6.3. Of course, if E is the K¨othe dual of a separable symmetric Banach n function space on R , then by Theorem 5.7 we may replace x M by + k=1 k E( ) k n ∞ (yk)k=1 E(M; ) in (34), where yk = i=1 xi. Appendix A. Original approach to Boyd’s theorem In this appendix we prove a full noncommutative analogue of Boyd’s interpola- tion theorem, which allows for interpolation of operators of weak type instead of only operators of Marcinkiewicz weak type. We adapt the original proof of Boyd [5]. The first result is an extension of Calder´on’s characterization of weak type op- erators ([6], Theorem 8). For 0 Proof. Suppose that q<∞. We may assume that max{Cp,Cq}≤1. Fix t>0, ∈ p,1 M q,1 M 1 − x ∞ let x L ( )+ + L ( )+ and set δ = μt(x). Define xt =(x δ)e (δ, )and 2 − 1 2 x x ∞ 1 2 ≥ ≥ xt = x xt ,soxt = xe [0,δ]+δe (δ, ). Observe that xt ,xt 0asx 0. Define two increasing, continuous functions φ1,φ2 : R+ → R+ by φ1(u)=(u − δ)χ{u>δ},φ2(u)=δχ{u>δ} + uχ{u≤δ}. From Proposition 4.1 it follows that 1 − μ(xt )=μ(φ1(x)) = φ1(μ(x)) = (μ(x) δ)χ{μ(x)≥δ}; 2 μ(xt )=μ(φ2(x)) = φ2(μ(x)) = δχ{μ(x)>δ} + μ(x)χ{μ(x)≤δ}. Using that δ = μt(x)weobtain 1 − μs(xt )=(μs(x) δ)χ[0,t](s)(s>0); 2 μs(xt )=δχ[0,t](s)+μs(x)χ(t,∞)(s)(s>0). 1 2 In particular, μ(x)=μ(xt )+μ(xt ). By midpoint convexity, ≤ 1 1 1 2 μt(Tx) μt( 2 T (2xt )+ 2 T (2xt )) ≤ 1 1 1 2 2 μ t (T (2xt )) + 2 μ t (T (2xt )). 2 2 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4108 SJOERD DIRKSEN By (38) we obtain t 1 − 1 1 − 1 1 1 ds 1 μ t (T (2x )) p t p x p,1 M = t p s p μs(x ) = Sp,qμ(x )(t) 2 t t L ( ) t t 0 s and, moreover, 2 − 1 2 μ t (T (2x )) q t q x Lq,1(M) 2 t t ∞ − 1 1 2 ds = t q s q μ (x ) s t s 0 t ∞ − 1 1 ds − 1 1 ds = t q s q δ + t q s q μs(x) . 0 s t s Since t t − 1 1 ds q − 1 1 ds t q s q δ = qδ = t p s p δ , 0 s p 0 s it follows that 2 2 μ t (T (2x )) p,q Sp,qμ(x )(t). 2 t t Putting our estimates together, we conclude that 1 2 μt(Tx) p,q Sp,qμ(xt )(t)+Sp,qμ(xt )(t)=Sp,qμ(x)(t). The case q = ∞ is proved similarly. The following observation for symmetric Banach function spaces and p, q ≥ 1 is the main result of [5]. The general case is proved using essentially the same argument (see [29], Theorem 2). Theorem A.2. If E is a symmetric quasi-Banach function space on R+, then the following hold. (a) If 0 Theorem A.3. Let E be a symmetric quasi-Banach function space on R+ which is s-convex for some 0 If p TxE(N ) p,q max{Cp,Cq}Sp,qE→E xE(M) (x ∈ E(M)+). Acknowledgement The author would like to thank the anonymous reviewer for comments on an earlier version of this paper and the resulting improvements. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NONCOMMUTATIVE BOYD INTERPOLATION THEOREMS 4109 References [1] Turdebek N. Bekjan and Zeqian Chen, Interpolation and Φ-moment inequalities of noncom- mutative martingales, Probab. Theory Related Fields 152 (2012), no. 1-2, 179–206, DOI 10.1007/s00440-010-0319-2. MR2875756 [2] T. Bekjan, Z. Chen, and A. Os¸ekowski, Noncommutative moment maximal inequalities. arXiv: 1108.2795 [3] A. Benedek, A.-P. Calder´on, and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 356–365. MR0133653 (24 #A3479) [4] Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathemat- ics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR928802 (89e:46001) [5] David W. Boyd, Indices of function spaces and their relationship to interpolation, Canad. J. Math. 21 (1969), 1245–1254. MR0412788 (54 #909) [6] A.-P. Calder´on, Spaces between L1 and L∞ and the theorem of Marcinkiewicz, Studia Math. 26 (1966), 273–299. MR0203444 (34 #3295) [7] Sjoerd Dirksen, Ben de Pagter, Denis Potapov, and Fedor Sukochev, Rosenthal inequalities in noncommutative symmetric spaces, J. Funct. Anal. 261 (2011), no. 10, 2890–2925, DOI 10.1016/j.jfa.2011.07.015. MR2832586 (2012k:46073) [8] S. Dirksen and E.´ Ricard, Some remarks on noncommutative Khintchine inequalities, Bull. London Math. Soc. 45 (2013), no. 3, 618–624. MR3065031 [9] Peter G. Dodds, Theresa K.-Y. Dodds, and Ben de Pagter, Noncommutative Banach func- tion spaces,Math.Z.201 (1989), no. 4, 583–597, DOI 10.1007/BF01215160. MR1004176 (90j:46054) [10] Peter G. Dodds, Theresa K. Dodds, and Ben de Pagter, Fully symmetric operator spaces, Integral Equations Operator Theory 15 (1992), no. 6, 942–972, DOI 10.1007/BF01203122. MR1188788 (94j:46062) [11] Peter G. Dodds, Theresa K.-Y. Dodds, and Ben de Pagter, Noncommutative K¨othe duality, Trans. Amer. Math. Soc. 339 (1993), no. 2, 717–750, DOI 10.2307/2154295. MR1113694 (94a:46093) [12] P. Dodds, B. de Pagter, and F. Sukochev, Noncommutative integration. Work in progress. [13] Thierry Fack and Hideki Kosaki, Generalized s-numbers of τ-measurable operators,Pacific J. Math. 123 (1986), no. 2, 269–300. MR840845 (87h:46122) [14] Ying Hu, Noncommutative extrapolation theorems and applications, Illinois J. Math. 53 (2009), no. 2, 463–482. MR2594639 (2011c:46137) [15] Yong Jiao, Burkholder’s inequalities in noncommutative Lorentz spaces, Proc. Amer. Math. Soc. 138 (2010), no. 7, 2431–2441, DOI 10.1090/S0002-9939-10-10267-6. MR2607873 (2011f:46080) [16] Yong Jiao, Martingale inequalities in noncommutative symmetric spaces,Arch.Math.(Basel) 98 (2012), no. 1, 87–97, DOI 10.1007/s00013-011-0343-1. MR2885535 [17] Marius Junge, Doob’s inequality for non-commutative martingales,J.ReineAngew.Math. 549 (2002), 149–190, DOI 10.1515/crll.2002.061. MR1916654 (2003k:46097) [18] Marius Junge and Quanhua Xu, Noncommutative Burkholder/Rosenthal inequalities, Ann. Probab. 31 (2003), no. 2, 948–995, DOI 10.1214/aop/1048516542. MR1964955 (2004f:46078) [19] Marius Junge and Quanhua Xu, Noncommutative maximal ergodic theorems,J.Amer. Math. Soc. 20 (2007), no. 2, 385–439, DOI 10.1090/S0894-0347-06-00533-9. MR2276775 (2007k:46109) [20] N. J. Kalton, Compact and strictly singular operators on certain function spaces,Arch.Math. (Basel) 43 (1984), no. 1, 66–78, DOI 10.1007/BF01193613. MR758342 (85m:47027) [21] N. J. Kalton, N. T. Peck, and James W. Roberts, An F -space sampler, London Mathe- matical Society Lecture Note Series, vol. 89, Cambridge University Press, Cambridge, 1984. MR808777 (87c:46002) [22] N. J. Kalton and F. A. Sukochev, Symmetric norms and spaces of operators,J.ReineAngew. Math. 621 (2008), 81–121, DOI 10.1515/CRELLE.2008.059. MR2431251 (2009i:46118) [23] S. Kre˘ın, Y. Petun¯ın, and E. Sem¨enov, Interpolation of linear operators,AmericanMathe- matical Society, Providence, R.I., 1982. MR0649411 (84j:46103) [24] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Function spaces. Ergeb- nisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. MR540367 (81c:46001) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4110 SJOERD DIRKSEN [25] Fran¸coise Lust-Piquard, In´egalit´es de Khintchine dans Cp (1 Hausdorff Center for Mathematics, Universitat¨ Bonn, Endenicher Allee 60, 53115 Bonn, Germany E-mail address: [email protected] Current address: Department of Mathematics, RWTH Aachen University, 52056 Aachen, Ger- many E-mail address: [email protected] License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use