ON THE RADEMACHER MAXIMAL FUNCTION MIKKOKEMPPAINEN Abstract. ThispaperstudiesanewmaximaloperatorintroducedbyHytönen,McIntoshand Portal in 2008 for functions taking values in a Banach space. The Lp-boundedness of this operator depends on the range space; certain requirements on type and cotype are present for instance. The original Euclidean definition of the maximal function is generalized to σ- finite measure spaces with filtrations and the Lp-boundedness is shown not to depend on the 1 underlying measure space or the filtration. Martingale techniques are applied to prove that a 1 weaktypeinequalityissufficientforLp-boundednessandalsotoprovideacharacterizationby 0 concavefunctions. 2 n Contents a J 1. Introduction 1 2 2. Preliminaries 2 2 3. The Rademacher maximal function 5 4. RMF-property, type and cotype 7 ] A 5. Reduction to Haar filtrations 9 6. The weak RMF-property 14 F . 7. RMF-property and concave functions 18 h Acknowledgements 22 t a References 22 m [ 3 1. Introduction v 8 The properties of the standard dyadic maximal function 5 3 Mf(ξ)= sup|(cid:104)f(cid:105) |, ξ ∈Rn, Q 3 Q(cid:51)ξ 2. where (cid:104)f(cid:105)Q denotes the average of a locally integrable function f over a dyadic cube Q, are well- 1 known. More precisely, the (sublinear) operator f (cid:55)→Mf is bounded in Lp for all p∈(1,∞] and 9 satisfiesforallf ∈L1 acertainweaktypeinequality(andalso,isboundedfromthedyadicHardy 0 space H1 to L1). These properties remain unchanged even if one studies functions taking values : v in a Banach space and replaces absolute values by norms. i In their paper [12], Hytönen, McIntosh and Portal needed a new maximal function in order X to prove a vector-valued version of Carleson’s embedding theorem. Instead of the supremum of r a (norms of) dyadic averages this maximal function measures their R-bound (see Section 2 for the definition), which in general is not comparable to the supremum. More precisely, they defined the Rademacher maximal function (cid:16) (cid:17) M f(ξ)=R (cid:104)f(cid:105) :Q(cid:51)ξ , ξ ∈Rn, R Q forfunctionsf takingvaluesinaBanachspace. TheyprovedthattheLp-boundednessoff (cid:55)→M f R is independent of p in the sense that boundedness for one p ∈ (1,∞) implies boundedness for all p in that range and that for many common range spaces including all UMD function lattices and 2010 Mathematics Subject Classification. Primary46E40;Secondary42B25,46B09. Key words and phrases. R-boundedness,martingales,typeandcotype. 1 2 M.KEMPPAINEN spaces with type 2, the operator M is Lp-bounded. Nevertheless it turned out that the new R maximal operator is not bounded for all choices of range spaces, e.g. not for (cid:96)1. ThestudyoftheRademachermaximaloperatorcontinueshereinabitmoregeneralframework, which was motivated by the need for vector-valued maximal function estimates in the context of non-homogeneous spaces in [11]. We consider it for operator-valued functions defined on σ-finite measurespaces,whereaveragesarereplacedbyconditionalexpectationswithrespecttofiltrations. The boundedness of M – the RMF-property (of the range space) – is shown not to depend on R these new parameters; instead, it is sufficient to check it for the filtration of dyadic intervals on [0,1) (Theorem 5.1). Here we follow a reduction argument from Maurey [17], originally tailored for the UMD-property. We also show that the RMF-property requires non-trivial type and finite cotype of the Banach spaces involved (Proposition 4.2). The Rademacher maximal function is readily defined for martingales X =(X )∞ of operators by j j=1 (cid:16) (cid:17) X∗ =R X :j ∈Z . R j + WewillshowusingideasfromBurkholder[4]thattheRMF-property(requiringLp-boundednessof M )isactuallyequivalent(Theorem6.6)totheweaktypeinequality(ortheweakRMF-property) R 1 P(X∗ >λ)(cid:46) (cid:107)X(cid:107) . R λ 1 Finally, the RMF-property is characterized using concave functions (Theorem 7.3) in the spirit of Burkholder [5]. 2. Preliminaries All random variables in Banach spaces (functions from a probability space to the Banach space) are assumed to be P-strongly measurable, by which we mean that they are P-almost everywhere limits of simple functions on the probability space whose measure we denote by P. Their expectation, denoted by E, is given by the Bochner integral. By an Lp-random variable, for 1≤p<∞, we mean a random variable X (in a Banach space) whose pth moment E(cid:107)X(cid:107)p is finite. Let (ε )∞ be a sequence of Rademacher variables, i.e. a sequence of independent random j j=1 variables attaining values −1 and 1 with an equal probability P(ε = −1) = P(ε = 1) = 1/2. j j By the independence we have E(ε ε ) = (Eε )(Eε ) = 0, whenever j (cid:54)= k, while (trivially) j k j k E(ε ε )=1, if j =k. The equality of a randomized norm and a square sum of norms for vectors j k x ,...,x in a Hilbert space is thus established by the following calculation: 1 N (cid:13)(cid:88)N (cid:13)2 (cid:68)(cid:88)N (cid:88)N (cid:69) (cid:88)N (cid:88)N (1) E(cid:13) ε x (cid:13) =E ε x , ε x = E(ε ε )(cid:104)x ,x (cid:105)= (cid:107)x (cid:107)2. (cid:13) j j(cid:13) j j k k j k j k j j=1 j=1 k=1 j,k=1 j=1 The following standard result guarantees the comparability of different randomized norms (see Kahane’s book [14] for a proof). Theorem 2.1. (TheKhintchine-Kahaneinequality)Forany1≤p,q <∞, thereexistsaconstant K such that p,q (cid:16) (cid:13)(cid:88)N (cid:13)p(cid:17)1/p (cid:16) (cid:13)(cid:88)N (cid:13)q(cid:17)1/q E(cid:13) ε x (cid:13) ≤K E(cid:13) ε x (cid:13) , (cid:13) j j(cid:13) p,q (cid:13) j j(cid:13) j=1 j=1 whenever x ,...,x are vectors in a Banach space. 1 N The concepts of type and cotype of a Banach space intend to measure how far the randomized norms are from (cid:96)p sums of norms. Definition. A Banach space E is said to have (1) type p for 1≤p≤2 if there exists a constant C such that (cid:16) (cid:13)(cid:88)N (cid:13)2(cid:17)1/2 (cid:16)(cid:88)N (cid:17)1/p E(cid:13) ε x (cid:13) ≤C (cid:107)x (cid:107)p (cid:13) j j(cid:13) j j=1 j=1 RADEMACHER MAXIMAL FUNCTION 3 for any vectors x ,...,x in E, regardless of N. 1 N (2) cotype q for 2≤q ≤∞ if there exists a constant C such that (cid:16)(cid:88)N (cid:17)1/q (cid:16) (cid:13)(cid:88)N (cid:13)2(cid:17)1/2 (cid:107)x (cid:107)q ≤C E(cid:13) ε x (cid:13) j (cid:13) j j(cid:13) j=1 j=1 for any vectors x ,...,x in E, regardless of N. In the case q =∞ the left hand side in 1 N the above inequality is replaced by max (cid:107)x (cid:107). 1≤j≤N j Remark. A few observations are in order. (1) As every Banach space has both type 1 and cotype ∞ we say that a Banach space has non-trivial type (respectively finite cotype) if it has type p for some p > 1 (respectively cotype q for some q <∞). (2) OnecanshowthatLp-spaceshavetypemin{p,2}andcotypemax{p,2}when1≤p<∞. Sequence spaces (cid:96)1 and (cid:96)∞ are, on the other hand, typical examples of spaces with only trivial type. (3) Type and cotype of a Banach space E and its dual E∗ are related in a natural way: If E has type p, then E∗ has cotype p(cid:48), where p(cid:48) is the Hölder conjugate of p. (4) The equality (1) of randomized norms and square sums of norms in Hilbert spaces means of course that they have both type 2 and cotype 2. A remarkable result of Kwapień’s (see the original paper [15], or the new proof by Pisier in [20]) is that a Banach space with both type 2 and cotype 2 is necessarily isomorphic to a Hilbert space. The geometry of a Banach space can be studied by looking at its finite dimensional subspaces. We denote by (cid:96)p , where p ∈ [1,∞] and N ∈ Z , the N-dimensional subspace of (cid:96)p consisting of N + sequences for which all but the first N terms are zero. A Banach space E is said to contain (cid:96)p ’s N λ-uniformly for a λ≥1 if there exist for each N ∈Z an N-dimensional subspace E of E and + N a bounded isomorphism Λ :E →(cid:96)p such that (cid:107)Λ (cid:107)(cid:107)Λ−1(cid:107)≤λ. N N N N N The following theorem of Maurey and Pisier (see [18] for the original proof, or [7], Theorems 13.3 and 14.1) relates this to the concept of type and cotype: Theorem 2.2. Suppose that E is a Banach space. Then (1) E has a non-trivial type if and only if it does not contain (cid:96)1 ’s uniformly (i.e. λ-uniformly N for some λ≥1). (2) E has finite cotype if and only if it does not contain (cid:96)∞’s uniformly. N Proposition 2.3. If E∗ has non-trivial type, then E has finite cotype. Proof. Non-trivial type implies finite cotype for the dual and thus it follows from the assumption that E∗∗ has finite cotype. By Theorem 2.2, E∗∗ does not contain (cid:96)∞’s uniformly and the same N has to hold for its subspace E. This means that E must have finite cotype. (cid:3) The proposition above, together with the fact that non-trivial type implies finite cotype, states in other words that if E has only infinite cotype, then both E and E∗ have only trivial type. Evidently, any infinite dimensional Hilbert space contains (cid:96)2 ’s 1-uniformly. The following N theoremisavariantofDvoretzky’stheorem(see[7],Theorems19.1and19.3ortheoriginalpaper by Dvoretzky [8]), which says that Banach spaces satisfy almost the same. The definition of K- convexity along with its fundamental properties can likewise be found in [7], Chapter 13. For the purposesofthispaper,onecanthinkofK-convexityasarequirementfornon-trivialtype. Indeed, aBanachspaceisK-convexifandonlyifithasnon-trivialtype([7],Theorem13.3). Furthermore, K-convexity is a self-dual property in the sense that a Banach space possesses it if and only if its dual does ([7], Corollary 13.7 and Theorem 13.5). Theorem 2.4. EveryinfinitedimensionalBanachspacecontains(cid:96)2 ’sλ-uniformlyforanyλ>1. N IftheBanachspaceisalsoK-convex,thenthereexistsaconstantC sothattheλ-isomorphiccopies of (cid:96)2 ’s can be chosen to be C-complemented. N 4 M.KEMPPAINEN We then turn to study the type of a space of operators. Suppose that H and E are Banach spaces. For y ∈E and x∗ ∈H∗ we write (y⊗x∗)x=(cid:104)x,x∗(cid:105)y, x∈H. Clearly y⊗x∗ ∈ L(H,E) and (cid:107)y⊗x∗(cid:107) ≤ (cid:107)y(cid:107)(cid:107)x∗(cid:107). We can also embed H∗ and E isometrically into L(H,E) by fixing respectively a unit vector y ∈ E or a functional x∗ ∈ H∗ with unit norm and writing H∗ (cid:39)y⊗H∗ :={y⊗x∗ :x∗ ∈H∗}⊂L(H,E) and E (cid:39)E⊗x∗ :={y⊗x∗ :y ∈E}⊂L(H,E). The following result is most likely well-known but in lack of reference we give a proof: Proposition 2.5. If H and E are infinite dimensional Banach spaces, then L(H,E) has only trivial type. Proof. Suppose first that H is K-convex and let λ > 1. By Theorem 2.4, both H and E contain (cid:96)2 ’s λ-uniformly. More precisely, thereexist sequences(H )∞ and (E )∞ ofsubspaces ofH N N N=1 N N=1 andE,suchthateachH andE isλ-isomorphicto(cid:96)2 . Now,asH isK-convex,wemayfurther N N N assume that for some constant C, each H is C-complemented in H so that the projection P N N onto H has norm less or equal to C. We can then embed L(H ,E ) in L(H,E) by extending N N N an operator T ∈ L(HN,EN) to T(cid:101) = TPN so that (cid:107)T(cid:101)(cid:107) ≤ C(cid:107)T(cid:107). Fix an N and denote the isomorphisms from H and E to (cid:96)2 by ΛH and ΛE, respectively. Define N N N N N Λ:L((cid:96)2 ,(cid:96)2 )→L(H ,E ) N N N N by Λ(T)=(ΛE)−1TΛH. Then Λ−1(S)=ΛES(ΛH)−1 and N N N N (cid:107)Λ(cid:107)(cid:107)Λ−1(cid:107)≤(cid:107)(ΛE)−1(cid:107)(cid:107)ΛH(cid:107)(cid:107)ΛE(cid:107)(cid:107)(ΛH)−1(cid:107)≤λ2. N N N N As every sequence in (cid:96)∞ defines a (diagonal) operator in L((cid:96)2 ,(cid:96)2 ) with same operator norm, N N N we have (cid:96)∞ (cid:44)→ L((cid:96)2 ,(cid:96)2 ) isometrically. Thus L(H,E) contains (cid:96)∞’s Cλ2-uniformly and cannot N N N N then by Theorem 2.2 have finite cotype, and thus cannot have non-trivial type either. Suppose then, that H is not K-convex. Then H∗ is not K-convex either, has only trivial type andcontains(cid:96)1 ’suniformly. ButH∗ (cid:44)→L(H,E)isometricallyandsoL(H,E)hasalsoonlytrivial N type. (cid:3) Inmanyquestionsofvector-valuedharmonicanalysistheuniformboundofafamilyofoperators has to be replaced by its R-bound (originally defined by Berkson and Gillespie in [2]). Definition. AfamilyT ofoperatorsinL(H,E)issaidtobeR-bounded ifthereexistsaconstant C such that for any T ,...,T ∈T and any x ,...,x ∈H, regardless of N, we have 1 N 1 N (cid:13)(cid:88)N (cid:13)p (cid:13)(cid:88)N (cid:13)p E(cid:13) ε T x (cid:13) ≤CpE(cid:13) ε x (cid:13) , (cid:13) j j j(cid:13) (cid:13) j j(cid:13) j=1 j=1 for some p ∈ [1,∞). The smallest such constant is denoted by R (T). We denote R by R in p 2 short later on. Basic properties of R-bounds can be found for instance in [6]. We wish only to remark that by the Khintchine-Kahane inequality, the R-boundedness of a family does not depend on p, and the constantsR (T)arecomparable. AsaconsequenceoftheinequalityR (T +S)≤R (T)+R (S) p p p p foranytwofamiliesT andS ofoperators,everysummablesequenceofoperatorsisalsoR-bounded: ∞ (cid:16) (cid:17) (cid:88) R {T }∞ ≤ (cid:107)T (cid:107). p j j=1 j j=1 WewillthencompareR-boundednessanduniformboundedness. AnyR-boundedsetisseento be uniformly bounded: sup(cid:107)T(cid:107) ≤R (T) L(H,E) p T∈T RADEMACHER MAXIMAL FUNCTION 5 for any 1≤p<∞. In Hilbert spaces also the converse holds. More generally, the following result is proven by Arendt and Bu in [1] (while the authors credit the proof to Pisier): Proposition 2.6. Suppose that H and E are Banach spaces. The following are equivalent: (1) H has cotype 2 and E has type 2. (2) Every uniformly bounded family of linear operators in L(H,E) is R-bounded. Remark. It is clear from above that if H and E have cotype 2 and type 2, respectively, and if X ⊂ L(H,E) is a Banach space whose norm dominates the operator norm, then all uniformly (X-) bounded sets are also R-bounded. ThereareatleasttwonaturalwaystouseR-boundednessforsetsofvectorsinE. Onecanfixa functionalx∗ withunitnormonaBanachspaceH andusetheembeddingE (cid:39)E⊗x∗ ⊂L(H,E). Doing so, a set S of vectors in E is R-bounded if there exists a constant C such that (cid:13)(cid:88)N (cid:13)p (cid:13)(cid:88)N (cid:13)p E(cid:13) ε (y ⊗x∗)x (cid:13) ≤CpE(cid:13) ε x (cid:13) (cid:13) j j j(cid:13) (cid:13) j j(cid:13) j=1 j=1 for any choice of vectors y ,...y ∈S and x ,...,x ∈H. 1 N 1 N In particular, one can choose the scalar field for H. As linear operators from the scalars to E are of the form λ(cid:55)→λy for some y ∈E, it makes sense to call a set S of vectors in E R-bounded if there exists a constant C such that (cid:13)(cid:88)N (cid:13)p (cid:12)(cid:88)N (cid:12)p E(cid:13) ε λ y (cid:13) ≤CpE(cid:12) ε λ (cid:12) (cid:13) j j j(cid:13) (cid:12) j j(cid:12) j=1 j=1 for all vectors y ,...,y in S and all scalars λ ,...,λ . These two conditions are easily seen to 1 N 1 N be equivalent. 3. The Rademacher maximal function SupposethatH andE areBanachspacesandthatX ⊂L(H,E)isaBanachspacewhosenorm dominates the operator norm. We are mostly interested in the case X (cid:39)E, i.e. when X =E⊗x∗ for some x∗ ∈H∗ or H is the scalar field. Another typical choice for X is L(H,E) itself. Further, when H is a Hilbert space, we can take the so-called γ-radonifying operators for our X (for the definition, seeLindeandPietsch[16], vanNeerven[19]orthebook[7]Chapter12). Theirnatural normisnotequivalenttotheoperatornorm,thusgivingusanon-trivialexampleofaninteresting X. Finally, for Hilbert spaces H and H one can consider the Schatten-von Neumann classes 1 2 S (H ,H ) with 1≤p<∞ (see [7] Chapter 4). p 1 2 We will now set out to define the Rademacher maximal function. Suppose that (Ω,F,µ) is a σ-finitemeasurespaceanddenotethecorrespondingLebesgue-BochnerspaceofF-measurableX- valuedfunctionsbyLp(F;X)(orLp(X)),1≤p≤∞. Thespaceofstronglymeasurablefunctions f for which 1 f is integrable for every set A∈F with finite measure, is denoted by L1(F;X). A σ If G is a sub-σ-algebra of F such that (Ω,G,µ) is σ-finite, there exists for every function f ∈L1(F;X) a conditional expectation E(f|G)∈L1(G;X) withrespect toG which isthe(almost σ σ everywhere) unique strongly G-measurable function satisfying (cid:90) (cid:90) E(f|G)dµ= fdµ A A foreveryA∈G withfinitemeasure. TheoperatorE(·|G)isacontractiveprojectionfromLp(F;X) onto Lp(G;X) for any p ∈ [1,∞]. This follows immediately, if the vector-valued conditional expectation is constructed as the tensor extension of the scalar-valued conditional expectation, which is a positive operator (see Stein [21] for the scalar-valued case). Conditional expectations satisfy Jensen’s inequality: If φ : X → R is a convex function and f ∈L1(X) is such that φ◦f ∈L1, then σ σ φ◦E(f|G)≤E(φ◦f|G) 6 M.KEMPPAINEN for any sub-σ-algebra G of F (for which (Ω,G,µ) is σ-finite). The proof in the case of a finite measure space can be found in [10]. Suppose then that (Fj)j∈Z is a filtration, that is, an increasing sequence of sub-σ-algebras of F such that each (Ω,F ,µ) is σ-finite. For a function f ∈ L1(F;X), we denote the conditional j σ expectations with respect to this filtration by E f :=E(f|F ), j ∈Z. j j The standard maximal function (with respect to (Fj)j∈Z) is given by Mf(ξ)=sup(cid:107)E f(ξ)(cid:107), ξ ∈Ω, j j∈Z for functions f in L1(X). The operator f (cid:55)→ Mf is known to be bounded from Lp(X) to Lp σ whenever 1<p≤∞, regardless of X. Definition. The Rademacher maximal function of a function f ∈L1(F;X) is defined by σ (cid:16) (cid:17) M f(ξ)=R E f(ξ):j ∈Z , ξ ∈Ω. R j Remark. Two immediate observations are listed below. (1) The µ-measurability of M f can be seen by studying it as the supremum over N of the R truncated versions (cid:16) (cid:17) M(N)f(ξ)=R E f(ξ):|j|≤N , ξ ∈Ω. R j Indeed, every M(N)f is a composition of a strongly µ-measurable function R Ω→X2N+1 :ξ (cid:55)→(E f(ξ))N j j=−N andacontinuousfunction(weassumedthatthenormofX dominatestheoperatornorm) (cid:16) (cid:17) X2N+1 →R: (T )N (cid:55)→R T :|j|≤N . j j=−N j (2) By the properties of R-bounds we obtain the pointwise relation Mf ≤ M f. If H has R cotype2andE hastype2itfollowsfromProposition2.6(andthefollowingremark)that M f (cid:46) Mf. This is the case in particular, when H = Lq for 1 ≤ q ≤ 2 and E = Lp for R 2≤p<∞ over some measure spaces. Example 3.1. Equip the Euclidean space Rn with the Borel σ-algebra and the Lebesgue measure. For each integer j, let D denote a partition of Rn into dyadic cubes with edges of length 2−j. j Suppose in addition, that every cube in D is a union of 2n cubes in D . For instance, one can j j+1 take the “standard” dyadic cubes Dj ={2−j([0,1)n+m):m∈Zn}. A filtration (Fj)j∈Z is then obtained by defining F as the σ-algebra generated by D . We write (cid:104)f(cid:105) for the average of an j j Q X-valued function f over a dyadic cube Q, that is 1 (cid:90) (cid:104)f(cid:105) = f(η)dη. Q |Q| Q Our maximal functions are now given by (cid:16) (cid:17) Mf(ξ)= sup(cid:107)(cid:104)f(cid:105) (cid:107) and M f(ξ)=R (cid:104)f(cid:105) :Q(cid:51)ξ , ξ ∈Rn. Q R Q Q(cid:51)ξ TheEuclideanversionoftheRademachermaximalfunctionwasoriginallystudiedbyHytönen, McIntosh and Portal [12] via the identification L(C,E) (cid:39) E. They showed using interpolation that the Lp-boundedness of f (cid:55)→ M f for one p ∈ (1,∞) implies boundedness for all p in that R range. They also provided an example of a space, namely (cid:96)1, for which the Rademacher maximal operator is not bounded. Definition. Let 1 < p < ∞. A Banach space X ⊂ L(H,E) is said to have RMF with respect p to a given filtration on a given σ-finite measure space if the corresponding Rademacher maximal operator is bounded from Lp(X) to Lp. RADEMACHER MAXIMAL FUNCTION 7 ThesmallestconstantforwhichtheboundednessholdswillbecalledtheRMF -constantforthe p givenfiltrationonthegivenmeasurespace. WhendealingwiththeEuclideancase,weoccasionally drop the subscript p and refer to the property as RMF with respect to Rn. Note that the RMF - p property is inherited by closed subspaces. In particular, if L(H,E) has RMF , then both E and p H∗ have it. WewillshowthatifX hasRMF withrespecttothefiltrationofdyadicintervalson[0,1),then p it has RMF with respect to any filtration on any σ-finite measure space. Supporting evidence is p found in the Euclidean case: If one restricts to the unit cube [0,1)n with the filtration of dyadic cubes contained in [0,1)n, it is not difficult to show that RMF with respect to this filtration on p [0,1)n is equivalent to RMF with respect to the filtration of standard dyadic cubes on Rn. p Martingales arelater onused tostudy aweaktype inequalityfor themaximal operator. In the Euclidean case, a similar inequality can be proven with the aid of Calderón-Zygmund decomposi- tion: Suppose that X ⊂ L(H,E) has RMF with respect to the filtration of dyadic cubes on Rn p for some p∈(1,∞), i.e. that M is bounded from Lp(X) to Lp. Then there exists a constant C R such that for all f ∈L1(X), C |{ξ ∈Rn :M f(ξ)>λ}|≤ (cid:107)f(cid:107) R λ L1(X) whenever λ > 0. The crucial part of the proof is to observe that M a vanishes outside a dyadic R cube containing the support of an atom a (whose average is zero). 4. RMF-property, type and cotype We will now study what kind of restrictions the boundedness of the Rademacher maximal operator puts on the type and cotype of the spaces involved. Unlike many other maximal operators, M is not in general bounded from L∞(L(H,E)) to R L∞. We actually have the following: Proposition4.1. TheRademachermaximaloperatorisboundedfromL∞(0,1;L(H,E))toL∞(0,1) if and only if H has cotype 2 and E has type 2. Proof. If H has cotype 2 and E has type 2, all the uniformly bounded sets are R-bounded and M f ≤CMf forallf inL∞(0,1;L(H,E)). Supposeonthecontrary,thatH doesnothavecotype R 2 or that E does not have type 2 and fix a C > 0. Now there exists a positive integer N and operators T ,...,T in L(H,E) with at most unit norm such that the R-bound of {T ,...,T } 1 N 1 N is greater than C. We then construct an L∞-function on [0,1) that obtains the operators T as j dyadic averages on an interval. Let us write I =[0,2j−N), j =1,...N, so that I =[0,21−N) is j 1 the smallest interval and I =[0,1). We set S =T and N 1 1 S =2T −T , j =2,...N. j j j−1 Now (cid:107)S (cid:107) ≤ 3 for all j = 1,...,N, so that if we define f(ξ) = S for ξ ∈ I and f(ξ) = S for j 1 1 j ξ ∈I \I , j =2,...,N, we have f ∈L∞(0,1;L(H,E)). j j−1 S S S S 1 2 3 4 I I \I I \I I \I 1 2 1 3 2 4 3 Figure 1. The construction of f with N =4 We then look at the averages of f over the intervals I . Obviously j (cid:104)f(cid:105) =S =T , I1 1 1 S +S T +2T −T (cid:104)f(cid:105) = 1 2 = 1 2 1 =T and I2 2 2 2 S +S +2S 2T +4T −2T (cid:104)f(cid:105) = 1 2 3 = 2 3 2 =T . I3 4 4 3 8 M.KEMPPAINEN More generally, observing the telescopic behaviour we calculate j 1 (cid:16) (cid:88) (cid:17) 1 (cid:104)f(cid:105) = S + 2k−1S = (T +2j−1T −T )=T , Ij 2j−1 1 k 2j−1 1 j 1 j k=1 for j = 2,...,N, as was desired. Thus M f > C on I , where C was chosen arbitrarily large R 1 and the bound 3 for the norm of f does not depend on C. The operator M cannot therefore be R bounded from L∞(0,1;L(H,E)) to L∞(0,1). (cid:3) Basedonthecounterexamplefrom[12]thatthesequencespace(cid:96)1 doesnothaveRMFweprove the following statement. Proposition 4.2. If for some p ∈ (1,∞), L(H,E) has RMF with respect to the usual dyadic p filtration on R, then H has finite cotype and E has non-trivial type. Proof. Suppose on the contrary that E has only trivial type. By Theorem 2.2 it follows that for someλ≥1thereexistsasequence(E )∞ ofsubspacesandasequence(ΛE)∞ ofisomorphisms N N=1 N N=1 between each E and (cid:96)1 such that (cid:107)ΛE(cid:107)(cid:107)(ΛE)−1(cid:107)≤λ. Let us then fix an N. It is shown in [12] N N N N that there exists a function f ∈Lp(0,1;(cid:96)1) for any p∈(1,∞) with the following properties: (1) f(ξ)∈(cid:96)1 for all ξ ∈[0,1), 2N (2) (cid:107)f(ξ)(cid:107)=1 for all ξ ∈[0,1) so that (cid:107)f(cid:107) =1, Lp(0,1;(cid:96)1) (3) (cid:107)M f(cid:107) ≥C loglogN, where the constant C does not depend on N. R Lp(0,1) 1 1 Define then a function g : [0,1) → E by g(ξ) = (ΛE )−1(f(ξ)) and note that (cid:107)g(cid:107) ≤ 2N Lp(0,1;E) (cid:107)(ΛE )−1(cid:107). Since M is bounded from Lp(0,1;E) to Lp(0,1) there exists a constant C such 2N R 2 that (cid:107)M g(cid:107) ≤ C (cid:107)g(cid:107) . But now, since f(ξ) = ΛE (g(ξ)) we have (cid:107)M f(ξ)(cid:107) ≤ R Lp(0,1) 2 Lp(0,1;E) 2N R (cid:107)ΛE (cid:107)(cid:107)M g(ξ)(cid:107). Thus 2N R (cid:107)M f(cid:107) ≤(cid:107)ΛE (cid:107)(cid:107)M g(cid:107) ≤C (cid:107)ΛE (cid:107)(cid:107)g(cid:107) ≤C λ R Lp(0,1) 2N R Lp(0,1) 2 2N Lp(0,1;E) 2 which gives a contradiction whenever N is chosen so large that C loglogN ≥C λ. 1 2 Theclaimonfinitecotypeisprovensimilarly. SupposeonthecontrarythatH hasonlyinfinite cotype. Then H∗ has only trivial type and one can proceed as above by defining a function h:[0,1)→H∗ by h(ξ)=ΛH∗(f(ξ)). (cid:3) 2N Recall that L(H,E) has only trivial type whenever H and E are infinite dimensional Banach spaces. Therefore it cannot have RMF via the identification L(H,E)(cid:39)L(C,L(H,E)). Since Lp-spaces have type 2 whenever 2≤p<∞, they have the RMF-property. We show next that they have RMF also when 1 < p < 2. This is implied by the following heredity property of RMF. Proposition 4.3. Let 1 < p < ∞. Suppose that (Σ,ν) is a σ-finite measure space and that X ⊂L(H,E)hasRMF withrespecttotheusualdyadicfiltrationonRn. ThenthespaceLp(Σ;X) p has RMF with respect to the usual dyadic filtration on Rn. p Proof. We use the identification Lp(Rn;Lp(Σ;X))(cid:39)Lp(Rn×Σ;X) and write (cid:16) 1 (cid:90) (cid:17) M(cid:102)Rf(ξ,η)=R |Q| f(ζ,η)dζ :Q(cid:51)ξ , (ξ,η)∈Rn×Σ, Q for the Rademacher maximal function in the first variable. By the RMF -property of X we have p for ν-almost every η that (cid:90) (cid:90) M(cid:102)Rf(ξ,η)pdξ (cid:46) (cid:107)f(ξ,η)(cid:107)pdξ. Rn Rn RADEMACHER MAXIMAL FUNCTION 9 We then calculate (cid:13)(cid:88) (cid:13)p (cid:90) (cid:12)(cid:88) 1 (cid:90) (cid:12)p E(cid:13) ε λ (cid:104)f(cid:105) (cid:13) = E(cid:12) ε λ f(ζ,η)dζ(cid:12) dν(η) (cid:13)Q(cid:51)ξ Q Q Q(cid:13)Lp(Σ;X) Σ (cid:12)Q(cid:51)ξ Q Q|Q| Q (cid:12) (cid:90) (cid:12)(cid:88) (cid:12)p (cid:46) M(cid:102)Rf(ξ,η)pdν(η)E(cid:12)(cid:12) εQλQ(cid:12)(cid:12) Σ Q(cid:51)ξ and so (cid:16) (cid:17)p (cid:90) R (cid:104)f(cid:105)Q :Q(cid:51)ξ (cid:46) M(cid:102)Rf(ξ,η)pdν(η). Σ Therefore, (cid:90) (cid:90) (cid:90) (cid:90) (cid:90) MRf(ξ)pdξ (cid:46) M(cid:102)Rf(ξ,η)pdξdν(η)(cid:46) (cid:107)f(ξ,η)(cid:107)pdξdν(η), Rn Σ Rn Σ Rn so that M is bounded from Lp((Lp(Σ;X)) to Lp. (cid:3) R Remark. ThepreviousPropositionfollowsalsofromthemoregeneralresultsprovenin[12],namely that both noncommutative Lp-spaces and all UMD function lattices have RMF. 5. Reduction to Haar filtrations WewillshowthattheRMF-propertyisindependentofthefiltrationandtheunderlyingmeasure space in the following sense: Theorem 5.1. Let 1 < p < ∞. If X has RMF with respect to the filtration of dyadic intervals p on [0,1), then it has RMF with respect to any filtration on any σ-finite measure space. p When this is the case, we simply say that X has RMF . The proof of Theorem 5.1 uses p ideas from Maurey [17], where a similar result is proven for the UMD-property. We begin with the simplest possible case of filtrations of finite algebras on finite measure spaces and proceed gradually toward more general situations. In order to do so, we first work on measure spaces (Ω,F,µ) with µ(Ω)=1, that are divisible in the sense that any set A∈F with positive measure has for all c∈(0,1) a (measurable) subset with measure cµ(A). By a basis of a finite subalgebra G of F we mean a partition of Ω into disjoint non-empty sets A ,...,A ∈ G that generate the subalgebra so that each A ∈ G can be expressed as a union of 1 m some of these A ’s. Such a partition, denoted by bsG, always exists and is unique. Observe that k functions measurable with respect to a finite algebra can be identified with functions defined on the basis of this algebra (or any finer algebra). Afiltration(F )∞ offinitesubalgebrasofF iscalledaHaar filtration ifbsF consistsofj+1 j j=1 j sets of positive measure. We also write F ={∅,Ω} so that bsF ={Ω}. Furthermore, every F 0 0 j is obtained from F by splitting a set B ∈bsF into two sets B(cid:48) and B(cid:48)(cid:48) of positive measure. j−1 j−1 A Haar filtration is said to be dyadic if in each splitting µ(B(cid:48)) (and hence also µ(B(cid:48)(cid:48))) are dyadic fractions of µ(B) and further to be standard if each B splits into sets of equal measure. A typical example of a filtration of finite algebras is of course the filtration of dyadic intervals on [0,1). We denote by D the finite algebra of dyadic intervals of length 2−j on [0,1) and so j bsD ={[(k−1)2−j,k2−j):k =1,...,2j}. j Supposethat(F )N isafiltrationoffinitealgebras. Byaddingonesetatatime(tothebasis), j j=1 one can construct a Haar filtration (F(cid:101)j)Kj=N1 that F(cid:101)1 ⊂F(cid:101)2 ⊂···⊂F(cid:101)K1 =F1 ⊂F(cid:101)K1+1 ⊂···⊂F(cid:101)KN =FN, where K +1 is the number of sets in bsF . Likewise, the filtration of dyadic intervals on [0,1) j j can be “embedded” in a standard Haar filtration on [0,1). 10 M.KEMPPAINEN bs F bs F 0 0 bs F bs F 1 1 bs F bs F 2 2 A filtration of finite algebras A Haar filtration bs F bs D 0 0 bs F bs D 1 1 bs F bs D 2 2 A standard Haar filtration The filtration of dyadic intervals Figure 2. Different filtrations of finite algebras Note that the RMF -constant of X with respect to a filtration (F )N of finite algebras is p j j=1 at least the RMF -constant with respect to any “subfiltration” (F )M , where 1 ≤ j ≤ ... ≤ p jk k=1 k1 j ≤N. Indeed, for any F -measurable f we have kM N (cid:16) (cid:17) (cid:16) (cid:17) R E(f|F )(A):1≤k ≤M ≤R E(f|F )(A):1≤j ≤N , A∈bsF , jk j N and the claim follows. Twofiltrations(Fj)∞j=1 and(F(cid:101)j)∞j=1 offinitealgebras(possiblyondifferentmeasurespaces)are said to be equivalent if there exists for every j ∈Z a measure preserving bijection between bsF + j andbsF(cid:101)j. ObservethatifbissuchabijectionfrombsFN tobsF(cid:101)N,thenforeveryFN-measurable f we have E(f|Fj)=E(f ◦b−1|F(cid:101)j)◦b for any j = 1,...,N. It is a matter of calculation that the RMF -constant of X (if finite) is the p same with respect to equivalent filtrations of finite algebras. Evidently, every filtration of finite algebras on any measure space (of total measure one) is equivalent to a filtration on the unit interval. The next lemma shows that when dealing with dyadic Haar filtrations, we can choose an equivalent filtration on the unit interval that very much resembles the filtration of dyadic intervals. The result goes back to Maurey [17] and a detailed proof can be found in Hytönen [10]. Lemma 5.2. Every dyadic Haar filtration on any measure space with total measure one is equiva- lent to a dyadic Haar filtration (F )N on the unit interval such that F ⊂D for some integers j j=1 j Kj K and j E(f|F )=E(f|D ), 1≤j ≤N, j Kj for any F -measurable f. N Hence, if X has RMF with respect to the filtration of dyadic intervals on [0,1), then it has p RMF with respect to any dyadic Haar filtration on any measure space with total measure one p and the RMF -constant is at most the RMF -constant with respect to the filtration of dyadic p p intervals. We say that X has RMF uniformly with respect to a class of filtrations on a class of measure p spaces if the RMF -constants in question are uniformly bounded. p For the next three lemmas, fix a divisible measure space (Ω,F,µ) with µ(Ω) = 1. In each of the lemmas we start with a filtration (F )∞ , truncate it at a positive integer N and construct a j j=1