Table Of Content

HARDY-LITTLEWOOD INEQUALITIES ON COMPACT QUANTUM GROUPS OF KAC TYPE SANG-GYUNYOUN 7 1 0 Abstract. TheHardy-LittlewoodinequalityonTcomparestheLp-normofa 2 functionwithaweightedℓp-normofitsFouriercoefficients. Theapproachhas recently been studied for compact homogeneous spaces and we study a natu- n ralanalogue intheframeworkofcompact quantum groups. Especially,inthe a J caseofthe reducedgroupC∗-algebrasandfreequantum groups, weestablish explicitLp−ℓp inequalitiesthroughinherentinformationofunderlyingquan- 1 tumgroup,suchasgrowthrateandrapiddecayproperty. Moreover,weshow 3 sharpness of the inequalities in a large class, including C(G) with compact A] gLrioeugprsouOpN+,C,Sr∗N(+G.)withpolynomiallygrowingdiscretegroupandfreequantum O . h t 1. Introduction a m Hardy and Littlewood [14] showed that, for each 1 < p 2, there exists a ≤ [ constant C such that p 1 1 v (1.1) ( (1+ n)2 p|f(n)|p)p1 ≤Cp||f ∼ f(n)zn||Lp(T) for all f ∈Lp(T). 2 n Z | | − n Z X∈ X∈ 92 This implies1that the mbultiplier Fw : Lp(T) →blp(Z), f 7→ (w(n)f(n))n∈Z, with 8 w(n) := is bounded. Moreover, this is a stronger form of the Sobolev 0 (1+ n)2−pp b 1. embedding the|o|rem 0 1 1 7 Hpp−q(T) Lq(T), 1<p<q < , ⊆ ∀ ∞ :1 where Hps(T):= f ∈Lp(T):(1−∆)s2(f)∈Lp(T) is the Bessel potential space. v “The Hardy-Littlewood inequality (1.1)” had been studied for compact abelian Xi groupsbyHewitt(cid:8)andRoss[15],andwasrecentlyext(cid:9)endedtocompacthomogeneous manifolds by Akylzhanov, Nursultanov and Ruzhansky [[1] and [2]]. For compact r a Lie groups G with the real dimension n, the result of [2] is written as follows: For each 10 such that p ≤ (1.2) (πX∈Gb (1+κπ1)n(22−p)nπ2−p2||f(π)||pHS)1p ≤Cp||f||Lp(G) for all f ∈Lp(G). b Here, G denotes a maximal family of mutually inequivalent irreducible unitary representations of G, A HS := tr(A∗A)12 and the Laplacian operator ∆ on G || || satisfies ∆b:π κ π for all π =(π ) G and all 1 i,j n . i,j 7→− π i,j i,j 1≤i,j≤nπ ∈ ≤ ≤ π The above inequality (1.2) can be reduced to a more familiar form whose left hand side is a natural weighted ℓp-norm of its Fourier cobefficients: 1 (1.3) (πX∈Gb (1+κπ)n(22−p)nπ||f(π)||pSnpπ)p1 ≤Cp||f||Lp(G). b Keywords and phrases. Hardy-Littlewoodinequality,Quantum groups,Fourieranalysis. 2010Mathematics Subject Classification. Primary20G42, Secondary46L52 Theauthor issupportedbyTJParkScienceFellowship. 1 Anotablepointisthat“theHardy-LittlewoodinequalitiesoncompactLiegroups (1.2)”aredeterminedbyinherentgeometricinformation,namelytherealdimension andthenaturallengthfunctiononG. Indeed,π √κπ isequivalenttothenatural 7→ length on G (See Remark 6.1). ||·|| The main purpose of this paperbis to establish new Hardy-Littlewood inequalities on compacbt quantum groups of Kac type through geometric information of underlying quantum groups. As a part of efforts, we will present some explicit inequalities in important examples. The reduced group C -algebras C (G) with ∗ r∗ discrete groups G, the free orthogonal quantum groups O+ and the free permu- N tation quantum groups S+ are the main targets. Of course, non-commutative Lp N analysis on quantum groups is widely studied from various perspectives ([7], [9], [16], [17] and [26]). For details of operator algebraic approach to quantum group itself, see [19], [20], [24] and [30]. To clarify our intention, let us show the main results of this article on “compact matrix quantum groups” which admit the natural length function : Irr(G) 0 N (See Definition 3.3 and Proposition 3.4). The following i|n·eq|ualities a→re { }∪ determined by inherent information of underlying quantum group, namely growth rate and rapid decay property. Theorem 1.1. Let G be a compact matrix quantum group of Kac type and denote by the natural length function on Irr(G). |·| (1) Let G have a polynomial growth with n2 (1+k)γ and γ >0. α ≤ α∈Irr(XG):|α|≤k Then, for each 1<p 2, there exists a universal constant K =K(p) such ≤ that 1 (1.4) (α∈XIrr(G)(1+|α|)(2−p)γnα||f(α)||pSnpα)p1 ≤( (1+ α1)(2 p)γn2α−p2b||f(α)||pHS)p1 ≤K||f||Lp(G) α∈XIrr(G) | | − b for all f n tr(f(α)uα) Lp(G). α ∼ ∈ (2) Let G haveαt∈hXeIrrr(aGp)id decaby property with universal constants C,β >0 such that f L∞(G) C(1+k)β f L2(G) || || ≤ || || forallf span( uα : α =k, 1 i,j n ). Defines := n2. ∈ i,j | | ≤ ≤ α k α (cid:8) (cid:9) α∈Irr(XG):|α|=k Then, for each 1<p 2, there exists a universal constant K =K(p) such ≤ that 1 (1.5) (kX≥0α∈Irr(XG):|α|=ksk(2−2p)(1+k)(2−p)(β+1)nα||f(α)||pSnpα)p1 ≤( (1+k)(21 p)(β+1)( nα||f(α)||b2HS)p2)1p ≤K||f||Lp(G) kX≥0 − α∈Irr(XG):|α|=k b for all f n tr(f(α)uα) Lp(G). α ∼ ∈ α∈XIrr(G) b Inparticular,itwasknownthattherapiddecaypropertyofF canbestrength- N ened in general holomorphic setting [18]. The improved result is called the strong Haagerup inequality. Based on this data, we can improve the Hardy-Littlewood inequality for C (F ) by focusing our attention on holomorphic forms. Theorem r∗ N 2 5.3 justifies the claim and it seems appropriate to call the improved one “strong Hardy-Littlewood inequality”. A natural view of the Hardy-Littlewood inequalities on compact Lie groups is that a properly chosen weight function w : G (0, ) makes the corresponding → ∞ multiplier :Lp(G) ℓp(G) given bybf (w(π)f(π)) Fw → 7→ π∈Gb be bounded for each 1 <p 2. Indeed, newly derived Hardy-Littlewood inequalities on compact quantum g≤roups wbill give a decay pair (rb,s) whose corresponding 1 multiplier is bounded, where w (α) := . Moreover, in Section Fwr,s r,s rα(1+ α)s 6,wewillshowthatthereisnoslowerdecaypair|s|uchth|a|t isboundedwhenG r,s F is one of the followings: C(G) with compact Lie groups, C (G) with polynomially r∗ growing discrete group and free quantum groups O+, S+. See Theorem 6.6. N N ThisapproachisquitenaturalbecauseitisstronglyrelatedtoSobolevembedding properties. Indeed, for the case G=Td, :Lp(Td) lp(Zd) is bounded if and Fw0,s → only if Hq2p−sp(q1−r1)(Td) Lr(Td) 1<q <r < , ⊆ ∀ ∞ where Hs(Td) is the Bessel potential space. p Lastly, in Section 7, we present some remarks that are by-products of this re- search. Weshowthatmostoffreequantumgroupsdonotadmitaninfinite(central) sidon set, and also give a Sobolev embedding theorem type interpretation of our results for C(G) with compact Lie group and C (G) with polynomially growing r∗ discrete group G. Also, we present an explicit inequality on quantum torus Td. θ 2. Preliminaries 2.1. Compact quantum groups. A compact quantum group G is given by a unital C -algebra A and a unital -homomorphism ∆:A A A satisfying ∗ min ∗ → ⊗ (1) (∆ id) ∆=(id ∆) ∆; ⊗ ◦ ⊗ ◦ (2)span ∆(a)(b 1 ): a,b A ,span ∆(a)(1 b): a,b A are dense in A. A A { ⊗ ∈ } { ⊗ ∈ } Every compact quantum group admits the unique Haar state h on A such that (h id)(∆(x))=h(x)1 =(id h)(∆(x)) for all x A. A ⊗ ⊗ ∈ AfinitedimensionalcorepresentationofGisgivenbyanelementu=(u ) i,j 1 i,j n n ≤ ≤ ∈ M (A) such that ∆(u ) = u u for all 1 i,j n. We say that the n i,j i,k k,j ⊗ ≤ ≤ k=1 X corepresentation u is unitary if u u = uu = Id 1 M (A) and irreducible if ∗ ∗ n A n X M :Xu=uX =C Id where Id is the⊗identi∈ty matrix in M . n n n n { ∈ } · Let uα =(uα ) be a maximal family of mutually inequivalent i,j 1≤i,j≤nα α Irr(G) finite dimensionalunitary irred∈uciblecorepresentationsofG. It is wellknownthat, (cid:8) (cid:9) for each α Irr(G), there is a unique positive invertible matrix Q M such ∈ α ∈ nα that tr(Q )=tr(Q 1) and α −α δ δ (Q 1) h((uβ ) uα )= α,β j,t −α i,s, α,β Irr(G),1 i,j n ,1 s,t n , s,t ∗ i,j tr(Q ) ∀ ∈ ≤ ≤ α ≤ ≤ β α δ δ (Q ) h(uβ (uα ) )= α,β i,s α j,t, α,β Irr(G),1 i,j n ,1 s,t n . s,t i,j ∗ tr(Q ) ∀ ∈ ≤ ≤ α ≤ ≤ β α We say that G is of Kac type if Q = Id M for all α Irr(G). In this α nα ∈ nα ∈ case, the Haar state h is tracial. Lastly,wedefineC (G)astheimageofAinthe GNSrepresentationoftheHaar r state h and L (G) := C (G) . The Haar state h has a normal faithful extension ∞ r ′′ to L (G). ∞ 3 2.2. Non-commutative Lp-spaces. Let be a von Neumann algebra with a M normal faithful tracial state ϕ. Note that the von Neumann algebra admits M the unique predual . We define L1(M,ϕ) := and L (M,ϕ) := , then ∞ M∗ M∗ M consider a contractive injection j : , given by [j(x)](y) := h(yx) for all M → M∗ y . The map j has dense range. ∈M Now ( , ) is a compatible pair of Banach spaces and we can define non- commutaMtiveMLp∗-space Lp( ,ϕ) := ( , ) for all 1 < p < , where (, ) is 1 1 M M M∗ p ∞ · · p the complex interpolation space. For any x L ( ,ϕ), its Lp-norm is given by ∞ ∈ M x Lp( ,ϕ) =ϕ(xp)p1 for all 1 p< . || |I|n pMarticular,|fo|r all 1 p ≤ , ∞we denote by Lp(G) the non-commutative Lp-space associated to the≤von≤Ne∞umann algebra L (G) of a Kac type compact ∞ quantum group G and the tracial Haar state h. Then the space of polynomials Pol(G):=span( uα :α Irr(G),1 i,j n ) i,j ∈ ≤ ≤ α is dense in Cr(G) and Lp(G) for(cid:8)all 1 p< . (cid:9) Under the assumption that G is of≤Kac ty∞pe, for 1 p< , ≤ ∞ ℓp(G):=(Aα)α∈Irr(G) ∈ Mnα : nαtr(|Aα|p)<∞  α∈YIrr(G) α∈XIrr(G)  b and the natural ℓp-norm of (Aα)α Irr(G) ℓp(G) is defined by  ∈ ∈ ||(Aα)α∈Irr(G)||ℓp(Gb) :=( nαtr(|Aαb|p))1p =( nα||Aα||pSnpα)p1. α∈XIrr(G) α∈XIrr(G) Also, ℓ∞(G):=(Aα)α∈Irr(G) ∈α∈YIrr(G)Mnα :α∈sIurrp(G)||Aα||<∞ b and the ℓ∞-norm of (Aα)α Irr(G) ℓ∞(G) is defined by  ∈ ∈ ||(Aα)α∈Irr(G)||ℓ∞b(Gb) :=αsIurrp(G)||Aα||. ∈ It is known that ℓ1(G)=(ℓ (G)) and ℓp(G)=(ℓ (G),ℓ1(G)) for all 1<p< ∞ ∞ 1 ∗ p . ∞ b b b b b 2.3. Fourier analysis on compact quantum groups. For a compact quantum group G, the Fourier transform :L1(G) ℓ (G), ψ ψ, is defined by ∞ F → 7→ (ψ(α)) :=ψ((uα ) ) for all α Irr(G),1 i,j n . i,j j,i ∗ ∈ b ≤b ≤ α It is also known that is an injective contractive map and it is an isometry from L2(G) ontbo l2(G) ([2F6], Proposition 3.1 and 3.2). Then, by the interpolation theorem, we induce the Hausdorff-Young inequality again, i.e. is a contractive map from Lp(G) intoblp′(G) for each 1 p 2, where p is the cFonjugate of p. ′ ≤ ≤ 2.4. The reduced group C -algebras. The reduced group C -algebra C (G), b ∗ ∗ r∗ can be defined for all locally compact groups, but we only consider discrete groups in this paper since we want to understand it as a compact quantum group. Definition 2.1. Let Gbe a discrete group and define λ B(l2(G)) for each g G g ∈ ∈ by [(λ )(f)](x):=f(g 1x) for all x G. g − ∈ Then the reduced group C -algebra, C (G), is defined as the norm-closure of the ∗ r∗ space span( λ : g G ) in B(l2(G)). Moreover, if we define a comultiplication g { ∈ } ∆ : C (G) C (G) C (G) by λ λ λ for all g G, then (C (G),∆) r∗ → r∗ ⊗min r∗ g 7→ g ⊗ g ∈ r∗ forms a compact quantum group. 4 Note that, for G=(C (G),∆) of a discrete group G, L (G) is nothing but the r∗ ∞ group von Neumann algebra VN(G) and Irr(G)= λ is identified with G. { g}g G ∈ 2.5. Free quantum groups of Kac type. Definition 2.2. (Free orthogonal quantum group [28]) Let N 2 and A be the universal unital C -algebra which is generated by the N2 ∗ ≥ self-adjoint elements u with 1 i,j N satisfying the relations: i,j ≤ ≤ N N u u = u u =δ for all 1 i,j N. k,i k,j i,k j,k i,j ≤ ≤ k=1 k=1 X X N Also, we define a comultiplication ∆ : A A A by u u u . min i,j i,k k,j → ⊗ 7→ ⊗ k=1 X Then (A,∆) forms a compact quantum group called the Free orthogonal quantum group. We denote it by O+. N Definition 2.3. (Free permutation quantum group [29]) Let N 2 and A be the universal unital C -algebra which is generated by the N2 ∗ ≥ self-adjoint elements u with 1 i,j N satisfying the relations: i,j ≤ ≤ N N u2i,j =ui,j =u∗i,j and ui,k = uk,j =1A for all 1≤i,j ≤N. k=1 k=1 X X N Also, we define a comultiplication ∆ : A A A by u u u . min i,j i,k k,j → ⊗ 7→ ⊗ k=1 X Then (A,∆) forms a compact quantum group called the Free permutation quantum group. We denote it by S+. N These free quantum groups are of Kac type, so that the Haar states are tracial states. Also, for all N 2, Irr(O+) and Irr(S+ ) can be identified with 0 N. ≥ N N+2 { }∪ Moreover, k+1 if G=O+ or S+ 2 4 n = k  2r0r−0Nr0k+1+ 2Nr0−−rN0 (N −r0)k+1 if G=ON+ or SN+2 with N ≥3 wherer isthelargestsolutionoftheequationX2 NX+1=0. Notethatn rk 0 − k ≈ 0 if N 3. ≥ 2.6. Thenoncommutative Marcinkiewiczinterpolationtheorem. Theclas- sicalMarcinkiewiczinterpolationtheoremhasanaturalnon-commutativeanalogue for semi-finite von Neumann algebras. Theorem 2.4. (The non-commutative Marcinkiewicz interpolation theorem [32]) Let M be equipped with a normal semifinite faithful trace φ and 1 p 0 such that for any T1 Lp1(M),T2 Lp2(M) and for any ∈ ∈ y >0, C (2.1) φ(1 (AT )) ( 1)p1 T p1 , (y,∞) | 1| ≤ y || 1||Lp1(M) C φ(1 (AT )) ( 2)p2 T p2 . (y,∞) | 2| ≤ y || 2||Lp2(M) Then A:Lp(M) Lp(N) is a bounded map. → Proof. The proof of [Theorem 1.22, [32]] is still valid under a natural modification. Also, the direct sum K := M N and natural extension A : K L1(K) gives ⊕ → another proof. (cid:3) 5 e If the sub-linear operator A satisfies the inequality (2.1), then we say that A is of weak type (p ,p ). Also, the boundedness of A:Lp(M) lp(N) implies that A 1 1 → is of weak type (p,p). Nowdenotebyc(Irr(G),ν)thespaceofallfunctionsonthediscretespaceIrr(G) with a positive measure ν. Then the above theorem is written as follows: Corollary 2.5. Let G be a Kac type compact quantum group and let 1 p < 1 p 0 such→that for any T1 Lp1(G),T2 Lp2(G) and ∈ ∈ for any y >0, C ν(α) ( 1)p1 T p1 , ≤ y || 1||Lp1(G) α: |(AXT1)(α)|≥y C ν(α) ( 2)p2 T p2 . ≤ y || 2||Lp2(G) α: |(AXT2)(α)|≥y Then A:Lp(G) ℓp(Irr(G),ν) is a bounded map. → 3. Paley-type inequalities 3.1. General Approach. Inthis subsection,wederiveaPaley-typeinequalityfor KactypecompactquantumgroupGviafundamentaltechniquessuchasHausdorff- Young inequality, Plancherel theorem and the non-commutative Marcinkiewicz interpolation theorem. We prove the following theorem by adapting techniques used in [2]. Theorem 3.1. Let G be a Kac type compact quantum group and let w :Irr(G) → (0, ) be a function such that C := sup t n2 < . Then, for each ∞ w  · α ∞ t>0  α: wX(α)≥t  10 such that ≤   (3.1) ( w(α)2−pn2α−p2||f(α)||pHS)1p ≤K||f||Lp(G) α∈XIrr(G) b for all f n tr(f(α)uα) Lp(G). α ∼ ∈ α∈XIrr(G) b Proof. Put ν(α) := w(α)2n2. We will show that the sub-linear operator A : α f(α) L1(G) → c(Irr(G),ν), f 7→ (|√| nαw||(HαS))α∈Irr(G), is a well-defined bounded map from Lp(G) into ℓp(Irr(G),ν) forball 10,since || √n|α| =( nα )2 ≤||f(α)||≤||f||L1(G), b b b ν(α) w(α)2n2 b ≤ α α: ||AfX(α)||HS≥y α: w(Xα)≤||fy||1 w(α)2 = n2dx α α: w(Xα) ||f||1 Z0 ≤ y (||f||1)2 y = [ n2]dx by the Fubini theorem α Z0 x12≤wX(α)≤||fy||1 ||f||1 y =2 t( n2)dt by substituting x to t2 α Z0 α: t≤wX(α)≤||fy||1 f 1 2C || || . w ≤ y This says that A is of weak type (1,1) with C =2C . 1 w Now, by Corollary 2.5, [ w(α)2−pnpα(p2−12)||f(α)||pHS]p1 .||f||Lp(G). α∈XIrr(G) b (cid:3) Thelefthandsideoftheinequality(3.1)canbereducedtoamorefamiliarform. Recall that the natural non-commutative ℓp-norm on ℓ∞(G)=ℓ∞−⊕α∈Irr(G)Mnα is given by ||(Aα)α∈Irr(G)||ℓp(Gb) =( nα||Aα||bpSnpα)p1 α∈XIrr(G) under the condition that G is of Kac type. Corollary 3.2. Let 1 < p 2 and w be a function satisfying the condition of ≤ Theorem 3.1. Then we have that ( w(α)2−pnα||f(α)||pSnpα))1p ≤K||f||Lp(G) α∈XIrr(G) b for all f n tr(f(α)uα) Lp(G). α ∼ ∈ α∈XIrr(G) b Proof. First, tr(f(α)p)= f(α) p . | | || ||Snpα 1 1 1 Put = . Then 2<r b and b r p − 2 ≤∞ tr(|f(α)|p)≤||f(α)||pHS||Idnα||pSnrα =nα1−p2||f(α)||pHS. This completes proof easily. (cid:3) b b b Nowwediscussanimportantsubclassofcompactquantumgroups,namelycom- pact matrix quantum groups which allows the natural length function on Irr(G). Definition 3.3. A compact matrix quantum group is given by a pair (A,∆,u) with a unital C -algebra A, a -homomorphism ∆ : A A A and a unitary u = ∗ min ∗ → ⊗ 7 n (u ) M (A) such that (1) ∆:u u u , (2) u=(u ) i,j 1≤i,j≤n ∈ n i,j 7→ i,k⊗ k,j ∗i,j 1≤i,j≤n k=1 X is invertible in M (A) and (3) u generates A as a C -algebra. n { i,j}1 i,j n ∗ ≤ ≤ Bydefinition,freeorthogonalquantumgroupsO+ andfreepermutationquantum N groups S+ are compact matrix quantum groups. Also, in the class of compact N quantum groups, the subclass of compact matrix quantum groups is characterized bythefollowingproposition. Theconjugateα Irr(G)ofα Irr(G)isdetermined ∈ ∈ by uα :=Qα12uαQα−21 =((Qα)i12,i(uαi,j)∗(Qα)j−,j21)1≤i,j≤nα. Proposition 3.4. ([24]) A compact quantum group is a compact matrix quantum group if and only if there exists a finite set S := α , ,α Irr(G) such that any α Irr(G) is 1 n { ··· } ⊆ ∈ contained in some iterated tensor product of elements α ,α , ,α ,α and the 1 1 n n ··· trivial corepresentation. Then there is a natural way to define a length function on Irr(G) ([25]). For non-trivial α Irr(G), the natural length α is defined by ∈ | | min m 0 N: β , ,β such that α β β , β α ,α n . { ∈{ }∪ ∃ 1 ··· m ⊆ 1⊗···⊗ m j ∈{ k k}k=1} The length of the trivial corepresentationis defined by 0. Thenwecanextractexplicitinequalities fromTheorem3.1andCorollary3.2by inserting geometric information of underlying quantum group, namely growth rate that is estimated by the quantities b := n2 [6]. k α α k |X|≤ Corollary 3.5. Let a Kac type compact matrix quantum group G satisfy b = n2 C(1+k)γ for all k 0 with C,γ >0 k α ≤ ≥ α Irr(G):α k ∈ X| |≤ with respect to the natural length function. Then, for each 1<p 2, there exists a ≤ universal constant K =K(p) such that 1 (3.2) (α∈XIrr(G)(1+|α|)(2−p)γnα||f(α)||pSnpα)p1 ≤( (1+ α1)(2 p)γn2α−p2b||f(α)||pHS)p1 ≤K||f||Lp(G) α∈XIrr(G) | | − b for all f n tr(f(α)uα) Lp(G). α ∼ ∈ α∈XIrr(G) b 1 Proof. Consider a weight function w(α):= . Then (1+ α)γ | | st>up0t·α:αXt−γ1 1n2α=0<sut≤p1t·α:αXt−γ1 1n2α≤C0<sut≤p1t·(t−γ1)γ =C. | |≤ − | |≤ − Now the conclusion isobtained by Theorem 3.1 andProposition3.2. (cid:3) 3.2. A paley-type inequality under the rapid decay property. In this subsection, we still assume that G is a compact matrix quantum group of Kac type. Oneofthemainobservationsofthispaperisthatthemoredetailedgeometricinfor- mation actually improves Theorem 3.1 and Corollary 3.2 in various “exponentially growing” cases. A more refined paley-type inequality can be obtained under the condition that G has the rapid decay property in the sense of [25]. 8 Definition 3.6. ([25]) Let G be a Kac type compact matrix quantum group. Then we say that G has the rapid decay property with respect to the natural length function on Irr(G) if there exist C,β >0 such that (3.3) nα nα || aαi,juαi,j||L∞(G) ≤C(1+k)β|| aαi,juαi,j||L2(G) α∈Irr(XG):|α|=kiX,j=1 α∈Irr(XG):|α|=kiX,j=1 for any k 0 and scalars aα C. ≥ i,j ∈ Notation 1. (1) When the natural length function on Irr(G) is given, we will use the notations S := α Irr(G): α =k and s := n2. (2) We denote by p tkhe or{tho∈gonal proj|ec|tion }from Lk2(G) toα∈thSek cαlousre of k P span( uα : α S , 1 i,j n ). i,j ∈ k ≤ ≤ α Proposition(cid:8)3.7. Suppose that a Kac typ(cid:9)e compact matrix quantum group G has the rapid decay property with respect to the natural length function on Irr(G) and with inequality (3.3). Then we have that (3.4) sup( α∈Irr(G):|α(k|=+kn1α)β||f(α)||2HS)12 ≤C||f||L1(G) for all f ∈L1(G). k 0 P ≥ b Proof. Since L1(G)is isometricallyembeddedinto the dualspaceM(G):=C (G) r ∗ and Pol(G) is dense in C (G), we have that r f L1(G) = sup f,x L1(G),L∞(G) || || x∈Pol(G):||x||L∞(G)≤1h i = sup f,x∗ L1(G),L∞(G) x∈Pol(G):||x||L∞(G)≤1h i = sup nαtr(f(α)x(α)∗) x∈Pol(G):||x||L∞(G)≤1α∈XIrr(G) (3.5) sup b b n tr(f(α)x(α) ) α ∗ ≥ x∈Pol(G):Pk≥0C(k+1)β(Pα:|α|=knα||xb(α)||2HS)12≤1α∈XIrr(G) nα b b sup sup tr(f(α)x(α)∗) ≥ C(k+1)β k≥0(Pα:|α|=knα||xb(α)||2HS)21≤1α∈Irr(XG):|α|=k =sup( α∈Irr(G):|α|=knα||f(α)||2HS)12. b b C(k+1)β k 0 P ≥ b (cid:3) Theorem 3.8. Let aKac type compact matrix quantumgroup G have the rapid de- caypropertywithrespecttothenaturallengthfunctiononIrr(G)andwithinequality (3.3). Also, suppose that a weight function w : 0 N (0, ) satisfies { }∪ → ∞ (3.6) C :=sup y (k+1)2β < . w   y>0 ·k 0:(Xk+1)β 1  ∞ ≥ w(k) ≤y Then, for each 10 such that ≤ (3.7) ( w(k)2−p( nα||f(α)||2HS)p2)p1 ≤K||f||Lp(G) kX≥0 α∈Irr(XG):|α|=k b for all f n tr(f(α)uα) Lp(G). α ∼ ∈ α∈XIrr(G) b 9 Proof. Put ν(k):=w(k)2. We will show that the sub-linear operator A:L1(G) → c( 0 N,ν), f (||pk(f)||L2(G)) , is a well-defined bounded map from Lp(G) k 0 { }∪ 7→ w(k) ≥ into lp( 0 N,ν) for all 10, ν(k) w(k)2 by Proposition 3.7. ≤ k≥0:(AXf)(k)>y k: w(k) X<C||f||L1(G) (k+1)β y w(k) Now put w(k):= . Then (k+1)β e w(k)2 = e (k+1)2βdx k:w(k)<XC||f||L1(G) Z0 e y (C||f||L1(G))2 y (k+1)2βdx ≤ Z0 k:√Xx≤we(k) C||f||L1(G) y =2 t (k+1)2βdt by substituting x=t2 · Z0 k:tX≤we(k) 2CwC f L1(G) || || . ≤ y Therefore, by Corollary 2.5, we obtain that ( w(k)2−p||pk(f)||pL2(G))p1 .||f||Lp(G). k 0 X≥ (cid:3) Corollary 3.9. LetaKactypecompact matrixquantumgroupGhavetherapid de- caypropertywithrespecttothenaturallengthfunctiononIrr(G)andwithinequality (3.3). Then, for each 1 0 ≤ such that (3.8) ( (1+k)(21 p)(β+1)( nα||f(α)||2HS)p2)1p ≤K||f||Lp(G) kX≥0 − α∈Irr(XG):|α|=k b for all f n tr(f(α)uα) Lp(G). α ∼ ∈ α∈XIrr(G) b 10