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NONCOMMUTATIVE ERGODIC AVERAGES OF BALLS AND SPHERES OVER EUCLIDEAN SPACES 7 1 GUIXIANGHONG 0 2 n a J Abstract. In this paper, we establish a noncommutative analogue of 8 Calderón’s transference principle, which allows us to deduce noncom- 1 mutativeergodicmaximalinequalitiesfromthespecialcase—operator- valued maximal inequalities. As applications, we deduce dimension- ] freeestimates ofnoncommutative Wiener’smaximalergodicinequality A andnoncommutative Stein-Calderón’smaximalergodicinequalityover Euclidean spaces. We also show the corresponding individual ergodic F theorems. ToshowWiener’spointwiseergodictheorem,weconstructa . h densesubsetonwhichpointwiseconvergenceholdsfollowingasomewhat t standardway. ToshowJones’pointwiseergodictheorem,weuseagain a transferenceprincipletogether withLittlewood-Paleymethod,whichis m differentfromJones’originalvariationalmethodthatisstillunavailable [ inthenoncommutative setting. 1 1. Introduction v 3 Let β (resp. σ ) be the normalized Lebesgue measure on the Euclidean 3 r r ball {v ∈ Rn; |v| ≤ r} (resp. the Euclidean sphere {v ∈ Rn : |v| = 9 ℓ2 ℓ2 4 r}). Let (X,m) be a standard measure space on which Rn acts measurably 0 by measure preserving transformation π. Let π(β ) (resp. π(σ )) denotes r r . the operator canonically associated to β (resp. σ ) on L (X). Wiener’s 1 r r p 0 pointwise ergodic theorem [32] asserts that π(βr)f(x) convergence to a limit 7 as r → ∞ for almost every x ∈ X provided f ∈ L (X) with 1 ≤ p < ∞. p 1 The limit is given by F(f), the projection of f to the fixed point subspace v: {g ∈Lp(X): π(v)g =g, ∀v ∈Rn}. While Jones’ pointwise ergodic theorem i [19] asserts π(σ )f(x) convergenceto F(f) as r →∞ for almost every x∈X X r providedf ∈L (X) with n/(n−1)<p<∞ and n>2 (See [21] for the case p r a n=2). The main tool used in the proof of pointwise convergence is maximal inequality. In both the ball and sphere cases, the ergodic maximal inequalities MR(2010) Subject Classification. Primary 46L52, 37A15; Secondary 46L51, 42B25. Keywords. Noncommutative Lp space, transference principle, ball and sphere averages, maximal ergodic theorem, individual ergodic theorem, spectral method. 1 2 GUIXIANGHONG are deduced, through Calderoń’s transference principle, from the corresponding maximalinequalities in the case X =Rn and the actiongivenby translation,thatis,fromtheHardy-Littlewoodmaximalinequality(intheballcase) and Stein’s spherical maximal inequality (in the sphere case). Once the er- godicmaximalinequalityis available,to showthe individualergodictheorem it suffices to identify a dense subset on which pointwise convergence holds. That the method in constructing a dense subset in the ball case is now standard. However, it is usually a difficult task to identify a dense subset in the sphere case since the spherical measure is singular. See Jones’ original proof [19]. The main purpose of this paper is to establish Wiener’s and Jones’ results in the noncommutative setting. That means, we are going to build maximal ergodic theorems and then pointwise ergodic theorems for general W∗-dynamical system (M,τ,Rn,α), where (M,τ) is a von Neumann algebra equipped with a trace τ and α : Rn → Aut(M) is a continuous trace-preserving group homomorphism (also called an action) in the weak ∗-topology. If we take for M the algebra L (X,m) with (X,m) a mea- ∞ sure space, τ the associatedintegraland α induced by an invertible measure- preserving transformation of X, we will recover Wiener’s and Jones’ results. For the main purpose, we first establish a noncommutative version of Calderoń’stransferenceprinciple. Calderoń’soriginalarguments do not work in the present setting, since there does not exist perfect notion of “point” on the noncommutative measure spaces. We overcome this difficulty using the ideas developed in the theory of vector-valued noncommutative L spaces. p The noncommutative transference principle reduce maximal ergodic inequalities to the operator-valued maximal inequalities, which have been shown in [23], [12]. It is worth to mention that in [23], Mei used noncommutative Doob’s inequality [15] to prove the Hardy-Littlewood maximal inequality on Rn, which yields that the bounds are of order O(2n). While in [12], the au- thorshowthattheboundscouldbetakentobeindependentofnbyadapting Stein-Strömberg’s idea [29], which is interesting in its own right. As in the classical setting, with maximal inequality at hand, to use density arguments to show the pointwise convergence, it suffices to find some dense subset such that pointwise convergence holds on it. In Wiener’s case, we construct a dense subset following a somewhat standard way, see Section 4 below. However, since sphere measures are singular, the dense subset con- tructed previously does not work in Jones’ case. On the other hand, Jones’ original variational method remains an open problem in the noncommutative setting. Our first attempt is via spectral method. We construct dense subsets on which pointwise convergence holds only when the dimension of Euclidean spaces n ≥ 4, see Remark 6.4. Motivated by Rubio de Francia’s 3 proof of Stein’s spherical maximal inequality (see for instance [26] [14]), we use Littlewood-Paleyfunction to decompose spherical means into pieces, and use transference principle to show each piece satisfies a generalized noncommutative Wiener’s ergodic theorem. Summing up all the pieces, we obtain noncommutative Jones’ ergodic theorem for all n≥3, see Section 5 below. As it is well-known that in the classical setting, Wiener’s ergodic theorem inspires many mathematicians to study the ball and sphere averaging prob- lems inergodictheory associatedto moregeneralgroups,see for instance the survey paper by Nevo [24]. We expect similar story would take place in the noncommutative setting. Actually, noncommutative ergodic theory has been developedsincetheverybeginningofthetheoryof“ringsofoperators”. How- everatthe earlystage,only mean ergodictheorems havebeen obtained. Itis until1976afterLance’spioneerwork[20]thatthestudy ofindividualergodic theorems really took off. Lance proved that the ergodic averages associated withanautomorphismofaσ-finitevonNeumannalgebrawhichleavesinvari- antanormalfaithfulstateconvergealmostuniformly. Lance’sworkmotivated somemathematicianstostudyindividualergodictheoremsassociatedtogen- eralgroups(seeforinstance[18]andreferencestherein). Alltheseresultscan be regarded as individual ergodic theorems in the case p = ∞, where maxi- malergodictheoremsholdtrivially. Onthe otherhand,Yeadon[33]obtained a maximal and pointwise ergodic theorem in the preduals of semifinite von Neumann algebras. Yeadon’s theorem provides a maximal ergodic inequality which might be understood as a weak type (1,1) inequality. In contrast with the classical theory, the noncommutative nature of these weak type (1,1) in- equalitiesseemsaprioriunsuitableforclassicalinterpolationarguments. The breakthrough was made in the previously quoted paper [18] by Junge and Xu. They established a sophisticated real interpolation method using well- established noncommutative L theory, which together with Yeadon’s weak p type (1,1) inequality allows them to obtain the noncommutative Dunford- Schwartz maximal ergodic theorem, thus the noncommutative individual ergodic theorem in L spaces for all 1 ≤ p < ∞. This breakthrough motivates p further reserach on noncommutative ergodic theorems including the present paper, see also [1] [13] [2] [10] [22] [11] and references therein. This paper is organized as follows. In the next section, we give some nec- essary preliminaries for formulating noncommutative ergodic theorems from the pointview of classical ergodic theory, such as noncommutative analogues of L spaces, maximal norms, pointwise convergence and measure-preserving p dynamical systems. In Section 3, we prove a noncommutative version of Calderoń’s transference principle. The noncommutative version of Wiener’s ergodic theorem is shown in Section 4. Section 5 is devoted to the proof of noncommutative Jones’ ergodic theorem. In the Appendix, we present the 4 GUIXIANGHONG spectral method to find a dense subset, which is particularly useful when the underlying group is not amenable. 2. Preliminaries and framework 2.1. Noncommutative L -spaces. Let (M,τ) be a noncommutative mea- p sure spaces, that is, M is a von Neumann algebra and τ is a normal semifinite faithful trace. Let S+ be the set of all positive element x in M with M τ(s(x)) < ∞, where s(x) is the smallest projection e such that exe = x. Let S be the linear span of S+ . Then any x∈S has finite trace, and S is M M M M a w∗-dense ∗-subalgebra of M. Let 1 ≤ p < ∞. For any x ∈ S , the operator |x|p belongs to S+ M M (|x|=(x∗x)12). We define 1 kxk = τ(|x|p) p, ∀x∈S . p M (cid:0) (cid:1) Onecancheckthatk·k iswelldefinedandisanormonS . Thecompletion p M of (S ,k·k ) is denoted by L (M) which is the usual noncommutative L - M p p p space associated with (M,τ). For convenience, we usually set L (M)=M ∞ equipped with the operatornormk·k . We refer the reader to [31] for more M information on noncommutative L -spaces. p 2.2. Noncommutative maximal norms. Letusrecallthedefinitionofthe noncommutativemaximalnormintroducedby Pisier[30]andJunge [15]. We define L (M;ℓ ) to be the space of all sequences x = (x ) in L (M) p ∞ n n≥1 p which admits a factorizationof the following form: there exist a,b∈L (M) 2p and a bounded sequence y =(y ) in L (M) such that n ∞ x =ay b, ∀n≥1. n n The norm of x in L (M;ℓ ) is given by p ∞ kxk =inf kak supky k kbk , Lp(ℓ∞) 2p n ∞ 2p (cid:8) n≥1 (cid:9) where the infimum runs over all factorizations of x as above. We will fol- low the convention adopted in [18] that kxk is sometimes denoted by Lp(ℓ∞) sup+x . n n p (cid:13) More g(cid:13)enerally,if Λ is any index set, we define L (M;ℓ (Λ)) as the space (cid:13) (cid:13) p ∞ of all x=(x ) in L (M) that can be factorized as λ λ∈Λ p x =ay b with a,b∈L (M), y ∈L (M), supky k <∞. λ λ 2p λ ∞ λ ∞ λ The norm of L (M;ℓ (Λ)) is defined by p ∞ sup+x = inf kak supky k kbk . (cid:13)λ∈Λ λ(cid:13)p xλ=ayλb(cid:8) 2p λ∈Λ λ ∞ 2p(cid:9) (cid:13) (cid:13) 5 It is shown in [18] that x∈L (M;ℓ (Λ)) if and only if p ∞ sup sup+x : J ⊂Λ, J finite <∞. λ p (cid:8)(cid:13)λ∈J (cid:13) (cid:9) (cid:13) (cid:13) Inthiscase, sup +x isequaltotheabovesupremum. Inthefollowing, λ∈Λ λ p we omit the(cid:13)index set Λ w(cid:13)hen it will not cause confusion. (cid:13) (cid:13) A closely relatedoperatorspace is L (M;ℓc ) for p≥2 which is the set of p ∞ all sequences (x ) ⊂L (M) such that n n p ksup+|x |2k1/2 <∞. n p/2 n≥1 While L (M;ℓr ) for p ≥ 2 is the Banach space of all sequences (x ) ⊂ p ∞ n n L (M) such that (x∗) ∈ L (M;ℓc ). All these spaces fall into the scope of p n n p ∞ amalgamatedL spacesintensivelystudiedin[17]. Thefollowinginterpolation p relationshipbetweensymmetricandasymmetricmaximalnormswillallowthe usage of square functions to control maximal norms in Section 5. Lemma 2.1. Let 2≤p≤∞. Then (Lp(M;ℓc∞),Lp(M;ℓr∞))12 =Lp(M;ℓ∞). Inadditionto the strongmaximalnormswhichcorrespondto L -normsof p maximalfunction,wearealsoconcernedwiththeweakmaximalnormswhich correspond to weak L -norms of maximal function. Given a sequence (x ) p n n in L (M), we define p k(xn)nkΛp(ℓ∞) =suλpλe∈Pin(fM){(τ(e⊥))p1 : kexnek∞ ≤λ}. When M is commutative, then infimum in this definition is attained at the projection 1 (sup |x |). Thus Λ (ℓ )-quasi norm is exactly the weak [0,λ] n n p ∞ L -quasi norm of maximal function. If we define the quasi Banach space p Λ (M;ℓ )tobethesetofallsequence(x ) ∈L (M)suchthatitsΛ (ℓ )- p ∞ n n p p ∞ quasinorm being finite, then these spaces have some nice interpolationprop- erties. We refer the readers to [18] for more information. Forgeneralindexset,Λ (ℓ )-quasinormandwhencequasiBanachspaces p ∞ can be defined similarly as in the definition of strong maximal norms. 2.3. Noncommutative pointwise convergence. Werecallanappropriate substitute for the usual almost everywhere convergence in the noncommuta- tivesetting. Thisis thealmostuniformconvergenceintroducedbyLance[20] (see also [18]). Let (x ) be a family ofelements inL (M). Recallthat (x ) is said λ λ∈Λ p λ λ∈Λ to converge almost uniformly to x, abbreviated by x → x,a.u if for every λ ε>0 there exists a projection e∈M such that τ(1−e)<ε and limke(x −x)k =0. λ ∞ λ 6 GUIXIANGHONG Also, (x ) is said to converge bilaterally almost uniformly to x, abbrevi- λ λ∈Λ ated by x → x,b.a.u if for every ε > 0 there is a projection e ∈ M such λ that τ(1−e)<ε and limke(x −x)ek =0. λ ∞ λ Obviously, if x → x,a.u, then x → x,b.a.u. On the other hand, in λ λ the commutative case, these two convergences are equivalent to the usual almost everywhere convergence in terms of Egorov’s theorem. However they are different in the noncommutative setting. As in[18], inordertodeduce thepointwiseconvergencetheoremsfromthe correspondingmaximalinequalities,itisconvenienttousetheclosedsubspace L (M;c ) of L (M;ℓ ). Recall that L (M;c ) is defined as the space of all p 0 p ∞ p 0 sequences (x ) ∈ L (M) such that there are a,b ∈ L (M) and (y ) ⊂ M n p 2p n verifying x =ay b and limky k =0. n n n ∞ n Similarly, for the study of the a.u convergence, we use the closed subspace L (M;cc) of L (M;ℓc ), which is defined to be the space of all sequences p 0 p ∞ (x )∈L (M) such that there are b∈L (M) and (y )⊂M verifying n p p n x =y b and limky k =0. n n n ∞ n The following lemma will be useful for our study of individual erogdic theorem (see [6]). Lemma 2.2. (i) If 1≤p<∞ and (x )∈L (M;c ), then x →0,b.a.u. n p 0 n (ii) If 2≤p<∞ and (x )∈L (M;cc), then x →0,a.u. n p 0 n 2.4. Group actions and relatednotions. Wewillcall(M,τ,Rn,α)aW∗- dynamicalsystemif(M,τ)isanoncommutativemeasurespaceandα: Rn → Aut(M)isacontinuoustrace-preservinggrouphomomorphism(alsocalledan action) in the weak ∗-topology. It is well-knownthat α is naturally extended to be isometric automorphisms of L (M) for all 1 ≤ p ≤ ∞, which is still p denoted by α. As it is well-known, the weak ∗-continuity of (α(u))u∈Rn on M induces the strongcontinuity of(α(u))u∈Rn, i.e. for eachx∈Lp(M) with 1 ≤ p < ∞, the map u → α(u)x is a continuous map from Rn to L (M), p where we take the norm topology on L (M). p Let F = {x ∈ M : α(u)x = x ,∀u ∈ Rn}. It is easy to show that F is a vonNeumannsubalgebraofMasinthesemigroupcase[18],thusthereexists a unique conditional expectation F : M → F. Moreover, this conditional expectation extends naturally from L (M) to L (F) with 1≤p<∞, which p p are still denoted by F. Let M(Rn) denote the Banach space of bounded complex Borel measures on Rn. For each µ∈M(Rn), there corresponds an operator α(µ), with norm 7 bounded by kµk in every L (M), 1≤p≤∞, given by 1 p α(µ)x= α(v)xdµ(v), ∀x∈L (M). Z p Rn This definition should be justified as follows. For any x ∈ L (M) and y ∈ p L (M)where1/p+1/q=1,thefunctionv →hα(v)x,yi iscontinuousonRn, q bounded by kxk kyk . Hence p q hα(v)x,yidµ(v) ≤kµk kxk kyk . Z 1 p q Rn It follows that the operator α(µ)x is well defined and α(µ)x is in L (M). p Moreover, kα(µ)xk ≤kµk kxk so that kα(µ)k≤kµk . p 1 p 1 It can be easily checked that tha map µ → α(µ) is a norm-continuous ∗-representation of the involutive Banach algebra M(Rn) as an algebra of operators on L (M). We recall that the product in M(Rn) is defined as 2 convolutionµ∗ν(f)= f(uv)dµ(u)dν(v),andtheinvolutionisµ∗(E)= Rn Rn µ(E)−1. R R Denote by P(Rn) the subset of probability measures in M(Rn). Let t → ν be a weakly continuous map from R to P(Rn), namely t → ν (f) is t + t continuous for each f ∈ C (Rn). We will refer to (ν ) as a one-parameter c t t>0 familyofprobabilitymeasures. Wecannowformulatethefollowingdefinition. Definition 2.3. A one-parameter family (ν ) ⊂ P(Rn) will be called a t t>0 global(resp. local)noncommutativepointwiseergodicfamilyinL ifforevery p W∗-dynamical system (M,τ,Rn,α) and every x ∈ L (M), α(ν )x converge p t bilaterally almost uniformly to F(x) (resp. x) as t tends to ∞ (resp. 0). 3. Noncommutative Calderoń’s transference principle In this section, we establish a noncommutative analogue of Calderoń’s transference principle. It is worth to mention that all the assertions in this sectionarestilltrueforgeneralamenablegroups,andthusageneralversionof noncommutativeCalderoń’stransferenceprincipleisstillavailable. Weprefer to prove them rigorously in our another work, since they are not raleted to the later applications in the present paper. Let λ be the action of Rn on the group itself by translation. Recall that an operator T is completely bounded on L (Rn) if for any noncommutative p measurespace(N,tr), T⊗id isboundedonL (L (Rn)⊗N);similarstate- N p ∞ ments hold for a sequence of operators or completely weakly bounded. See [30] for detailed informations. Theorem 3.1. Let (ν ) be a family of probability measures on Rn having t t>0 theirsupportcontainedinaEuclidean ballofradius CtwhereC isan absolute constant. If the family of operators (λ(ν )) were completely bounded from t t>0 8 GUIXIANGHONG L (Rn) to L (L (Rn);ℓ ) (resp. to Λ (L (Rn);ℓ )), then for any W∗- p p ∞ ∞ p ∞ ∞ dynamical system (M,τ,Rn,α), the family of operators (α(ν )) is com- t t>0 pletely bounded from L (M) to L (M;ℓ ) (resp. to Λ (M;ℓ )). p p ∞ p ∞ In the rest of this section, we fix a W∗-dynamical system (M,τ,Rn,α). The following three lemmas arewell-knownin the commutative case,but not obvious in the noncommutative setting. Lemma 3.2. Let 1≤p≤∞. Then for (x ) in L (M), we have t t>0 p k(x ) k =supk(x ) k t t>0 Lp(ℓ∞) t 0<t≤T Lp(ℓ∞) T and further we have k(x ) k ≃supk(x ) k t t>0 Λp(ℓ∞) t 0<t≤T Λp(ℓ∞) T for positive x ’s. t Proof. The first equation has been proved in [18] using the duality between Lp′(M;ℓ1) and Lp(M;ℓ∞). Let us prove the second equivalence. The left handsidecontrolstriviallytherighthandside. Non-trivialpartisthereverse inequality. We start with the observation {e∈P(M):kex ek ≤λ, ∀t>0} t ∞ = {e∈P(M):kex ek ≤λ, ∀0<t≤T}. t ∞ \ T By density we may assume that (x ) ∈ ℓ (S ). Now, given λ > 0 and t t>0 ∞ M T >0, by definition, there exists a projection e ∈M such that T ke x e k ≤λ T t T ∞ for any t∈(0,T], and the right hand side dominates modulo a constant 1 λτ 1−e p. T (cid:0) (cid:1) Defineu=w∗−L −lim e . Recallthatuisnotnecessarilyaprojection. ∞ σ,U T However,recalling that (x ) ∈ℓ (S ), it is straightforwardto show that t t>0 ∞ M the exactsame inequalities above apply for u insteadof e , details are left to T the reader. Then, the projection e=χ[12,1](u) clearly satisfies e≤2eue≤4u2 and 1−e≤2(1−u). 1 This implies for any t ∈ (0,∞) that ex e ≤ 2 y∗u2y 2 ≤ 2λ where y t ∞ t t ∞ t 1(cid:13) (cid:13)1 (cid:13) 1 (cid:13) such that xt = yt∗yt, and λτ 1−e p(cid:13)≤ 2(cid:13)pλτ 1−(cid:13)u p. T(cid:13)hus we finish the proof. (cid:0) (cid:1) (cid:0) (cid:1) (cid:3) 9 Lemma 3.3. Let 1 ≤ p ≤ ∞. Then for any v ∈ Rn and (x ) in L (M), t t>0 p we have k(x ) k =k(α(v)x ) k t t>0 Lp(ℓ∞) t t>0 Lp(ℓ∞) and k(x ) k =k(α(v)x ) k . t t>0 Λp(ℓ∞) t t>0 Λp(ℓ∞) Proof. We show the first equality. By the definition of L (ℓ )-norm, for any p ∞ ε>0, there exist a factorization x =ay b such that t t ε+k(x ) k ≥kak supky k kbk . t t>0 Lp(ℓ∞) 2p t ∞ 2p t>0 Since α is a homomorphism, we find a factorization of α(v)x , t α(v)x =(α(v)a)(α(v)y )(α(v)b) t t with kα(v)ak supkα(v)y k kα(v)bk =kak supky k kbk . 2p t ∞ 2p 2p t ∞ 2p t>0 t>0 Since ε is arbitrary, we obtain k(x ) k ≥k(α(v)x ) k . t t>0 Lp(ℓ∞) t t>0 Lp(ℓ∞) The reverse inequality is shown similarly. The second equality follows from the following observation inf {(τ(e⊥))p1 : keα(v)xtek∞ ≤λ} e∈P(M) = inf {(τ((α(v−1)e)⊥))p1 : kα(v)(α(v−1)(e)xtα(v−1)(e))k∞ ≤λ} e∈P(M) = inf {(τ((α(v−1)e)⊥))p1 : k(α(v−1)(e)xtα(v−1)(e)k∞ ≤λ}. α(v−1)e∈P(M) where we have used the facts α(v−1)e⊥ = (α(v−1)e)⊥ and α is a trace- preserving homomorphism. (cid:3) Lemma 3.4. Let 1≤p≤∞ and (f ) ⊂L (L (Rn)⊗¯M). Then we have t t>0 p ∞ (ZRnk(ft(v))t>0kpLp(M;ℓ∞)dv)p1 ≤k(ft)t>0kLp(L∞(Rn)⊗¯M;ℓ∞) and inf {τ(e⊥): ke f (v)e k ≤λ}dv ZRnev∈P(M) v v t v ∞ ≤ inf {τ (e⊥): kef ek ≤λ}. e∈P(L∞(Rn)⊗M) Z t ∞ 10 GUIXIANGHONG Proof. Bydefinition, foranyε>0,there existsa factorizationf =ag b such t t that kak supkg k kbk ≤k(f ) k +ε. 2p t ∞ 2p t t>0 Lp(ℓ∞) t We see that for each v ∈ Rn, we find a factorization f (v) = a(v)g (v)b(v) t t such that the left hand side is smaller than (Z ka(v)kp2psupkgt(v)kp∞kb(v)kp2pdv)p1 ≤kak2psupkgtk∞kbk2p. Rn t t Since ε is arbitrary, we obtain the desired result. Nowweshowthesecondinequality. Foranyε>0,thereexistsaprojection e such that ε τ (e⊥)≤ inf {τ (e⊥): kef ek ≤λ}+ε. Z ε e∈P(L∞(Rn)⊗M) Z t ∞ Observing that for each v ∈Rn, we find a projection e (v) such that ε ke (v)f (v)e (v)k ≤λ, ε t ε ∞ so the left hand side is controlled by τ(e (v)⊥)dv Z ε Rn which yields the desired estimate since ε is arbitrary. (cid:3) Now we are at a position to prove Theorem 3.1. Proof. The case of the strong type (p,p). By Lemma 3.2, it suffices to prove for any T >0 (3.1) k(α(ν )x) kp ≤Cpkxkp t 0<t≤T Lp(ℓ∞) p p for any x∈L (M) where C is a constant independent of x and T. p p GivenanyS >0,byLemma3.3andLemma3.4,thelefthandsideof(3.1) equals 1 k(α(v)α(ν )x) kp dv |BS|ZBS t 0<t≤T Lp(M;ℓ∞) 1 ≤ k(χ (v)α(v)α(ν )x) kp . |BS| BS t 0<t≤T Lp(L∞(Rn)⊗M;ℓ∞) Define the L (M)-valued function f on Rn by f(u) = χ (u)α(u)x. p |u|≤S+CT Then for 0<t≤T, it is easy to check χ (v)α(v)α(ν )x=χ (v)λ⊗¯id (ν )f(v). BS t BS M t