Sobolev embedding properties on compact matrix quantum groups of Kac type

We establish sharp Sobolev embedding properties within a broad class of compact matrix quantum groups of Kac type under the polynomial growth or the rapid decay property of their duals. Main examples are duals of polynomially growing discrete quantum groups, duals of free groups and free quantum groups $O_N^+,S_N^+$. In addition, we generalize sharpend Hausdorff-Young inequalities, compute degrees of the rapid decay property for $\widehat{O_N^+},\widehat{S_N^+}$ and prove sharpness of Hardy-Littlewood inequalities on duals of free groups.


INTRODUCTION
It is a long tradition to study Fourier multipliers in harmonic analysis, and L p´Lq multipliers have played major roles in the theory of partial differential equations. A representative example is the Sobolev embedding property. More precisely, the Hardy-Littlewood-Sobolev theorem on tori states that (1.1) for all 1 ă p ă q ă 8 and f P L p pT d q, where ∆ is the Laplacian operator. Sobolev embedding properties have been explored in a broad class including Lie groups [Fol75,FR17,Var88,BPTV18], and more generally, L p´Lq multipliers on Lie groups have been extensively studied [CGM93,AR15,ANR15,ANR16]. In particular, for connected compact Lie groups G, it is known that for any 1 ă p ď 2 (1.2) ÿ πPIrrpGq n π p1`κ π q np 1 where n is the real dimension of G, ∆ : π i,j Þ Ñ´κ π π i,j is the Laplacian operator and }A} HS " trpA˚Aq 1 2 . Since the natural length function |¨| on IrrpGq satisfies |π| « κ 1 2 π [Wal73, Lemma 5.6.6], the above (1.2) is equivalent to Note that (1.3) detects the real dimension of G, which is an important geometric quantity. The main purpose of this study is to generalize (1.3) to the framework of compact quantum groups by employing geometric information of the underlying quantum group such as growth rates and the rapid decay property. Indeed, for highly important examples, we will show that the polynomial growth order or the degree of rapid decay property replaces the role of the real dimension n.
The theory needed to explore Sobolev embedding properties for compact quantum groups is so-called non-commutative L p -analysis. On quantum groups and quantum tori, L pĹ q multipliers have been studied from various perspectives [JPPP17, GPJP17, FHL`17, AMR18,XXY18]. In particular, due to [AMR18,Theorem4.3] which generalizes a theorem of H: ormander, if G is a compact matrix quantum group of Kac type whose dual p G has a polynomial growth (1.4) b k " ÿ αPIrrpGq:|α|ďk n 2 α ď Cp1`kq γ for some C, γ ą 0, then for any 1 ă p ď 2 we have where |¨| is the natural length function on IrrpGq. To our best knowledge, if we exclude compact Lie groups and duals of polynomially growing discrete groups, it is not known whether the above inequalities (1.5) are sharp. However, we will show that (1.5) is sharp if one of the following natural assumptions holds: (1) (Corollary 3.4) if b k « p1`kq γ and there exists a standard noncommutative semigroup pT t q tą0 on L 8 pGq whose infinitesimal generator L satisfies (1.6) Lpu α i,j q "´lpαqu α i,j with lpαq " |α|.
The above results establish sharp Sobolev embedding properties for connected compact Lie groups, duals of polynomially growing discrete groups, O2 and S4 .
Despite the above strong conclusion under the polynomial growth of p G, it is important to focus on duals of free groups x F N and free quantum groups OǸ , SǸ whose duals are exponentially growing. Arguably, these are the most important examples of compact quantum groups in view of operator algebras [Wan95, VDW96, Wan98, Voi11, Bra12, VV13, Iso15, FV16, BCV17, Iso17, BV18], and non-commutative L p -analysis on x F N , OǸ , SǸ has been extensively studied [JMP14b, MR17, MdlS17, JPPP17, Wan17, FHL`17, You18b,You18a]. From this viewpoint, one of the main aims of this paper is to establish the analogues of (1.5) for x F N , OǸ , SǸ sharply.
To settle this, we will take two strategies. First of all, we extend [You18a, Theorem 3.2] to genral compact matrix quantum groups whose duals have the rapid decay property (Theorem 4.1). We call it sharpened Hausdorff-Young inequalities and explain why such a phenomenon does not appear in the category of compact Lie groups, duals of discrete groups, O2 and SU q p2q (Section 4.1). Then, by applying the complex interpolation method between the sharpened Hausdorff-Young inequalities (Theorem 4.1) and Hardy-Littlewood inequalities [You18b, Theorem 3.8], for any 1 ă p ď 2 we obtain under the assumption that G is one of x F N , OǸ`1 and SǸ`3 with N ě 2. Secondly, the problem to check whether the exponent 3 in (1.7) is optimal amounts to ultracontractivity problems of certain semigroups associated with free groups or free quantum groups due to [Xio17, Theorem 1.1]. More precisely, for the Poisson or heat semigroup pT t q tą0 of G " x F N , OǸ`1 or SǸ`3, we will prove that there exists a universal constant K ą 0 such that for all f P L 1 pGq and t ą 0 2 if and only if d ě 3 (Corollary 6.2). This confirms that (1.7) is sharp for x F N , OǸ`1 and SǸ`3 with N ě 2.
Lastly, we note that this study is applicable (1) (Corollary 5.4) to calculate the rapid decay degrees of y OǸ and x SǸ and (2) (Corollary 6.3) to prove sharpness of Hardy-Littlewood inequalities on x F N presented in [You18b,Theorem 4.4].
2. PRELIMINARIES 2.1. Compact quantum groups and the representation theory. A compact quantum group G is a pair pCpGq, ∆q where CpGq is a unital C˚-algebra and ∆ : CpGq Ñ CpGq b min CpGq is a unital˚-homomorphism such that (1) p∆ b idq˝∆ " pid b ∆q˝∆.
(2) span t∆paqpb b 1q : a, b P CpGqu and span t∆paqp1 b bq : a, b P CpGqu are dense in CpGq b min CpGq. For a compact quantum group G, there exists a unique state h satisfying We call h the Haar state and G is said to be of Kac type if h is tracial.
We say that a unitary representation u is irreducible if tT P M nu : pT b 1qu " upT b 1qu " C¨Id nu and denote by IrrpGqu α " pu α i,j q 1ďi,jďnα PolpGq " span u α i,j : α P IrrpGq and1 ď i, j ď n α ( is a dense˚-subalgebra of CpGq and the Haar state h is faithful on PolpGq. Associated to a compact quantum group G is the discrete dual quantum group p G " pℓ 8 p p Gq, p ∆, p hq. Among the structures of p G, the underlying von Neumann algebra ℓ 8 p p Gq is ℓ 8´' αPIrrpGq M nα and, if G is of Kac type, p h is a normal semifinite faithful tracial weight on ℓ 8 p p Gq given by See [Wor87b, Wor87a, KV00, KV03, Tim08] for more details of locally compact quantum groups.
2.2. Non-commutative L p -spaces. Throughout this paper, we assume that G is a compact quantum group of Kac type. Since h is faithful on PolpGq, the space PolpGq is canonically embedded into BpL 2 pGqq where L 2 pGq is the completion of PolpGq with respect to the inner product xf, gy L 2 pGq " hpg˚f q for all f, g P PolpGq.
We define an associated von Neumann algebra L 8 pGq " PolpGq 2 in BpL 2 pGqq, and then the Haar state h extends to a normal faithful tracial state h on L 8 pGq.
For any 1 ď p ă 8, the non-commutative L p -space is defined as the completion of PolpGq with respect to the norm structure }f } L p pGq " hp|f | p q 1 p for any f P PolpGq. Then it is well known that pL 8 pGq, L 1 pGqq 1 p " L p pGq where p¨,¨q θ is the complex interpolation space and that L p 1 pGq " L p pGq˚for any 1 ď p ă 8 under the dual bracket xf, gy L p pGq,L p 1 pGq " hpgf q for all f, g P PolpGq.
On the dual side, for any 1 ď p ă 8, the non-commutative ℓ p -space is defined as , .
and the natural norm structure is }A} ℓ p p p Gq "¨ÿ αPIrrpGq n α trp|A α | p q‚ 1 p for all A P ℓ p p p Gq.
Then pℓ 8 p p Gq, ℓ 1 p p Gqq 1 p " ℓ p pGq and ℓ p 1 p p Gq " ℓ p p p Gq˚hold for any 1 ď p ă 8. The dual bracket between ℓ p p p Gq and ℓ p 1 p p Gq is given by For the general theory of non-commutative L p -spaces, see [PX03,Pis03].
2.3. Compact matrix quantum groups and the rapid decay property. The tensor product representation of two unitary representations u and v is and every unitary representation is decomposed into a direct sum of irreducible unitary representations. If σ is an irreducible component of u J v, we write σ Ď u J v. A compact matrix quantum group is a compact quantum group G for which there exists a unitary representation w such that every u α P IrrpGq is an irreducible component of w J n for some n P t0u Y N. In this case, we can define a natural length function on IrrpGq by (2.9) |α| " min for all α P IrrpGq.
Throughout this paper, we will use the following notations frequently.
Notation 1. Let G be a compact matrix quantum group and k P t0u Y N.
(1) k-balls are defined as B k " tα P IrrpGq : |α| ď ku and b k is defined by (2) k-spheres S k are defined as tα P IrrpGq : |α| " ku and s k is defined by (3) For each α P IrrpGq, we define H α " span u α i,j : 1 ď i, j ď n α ( and denote by p α the orthogonal projection from L 2 pGq onto H α .
We say that p G has a polynomial growth if there exists C, γ ą 0 such that b k ď Cp1`kq γ [BV09] and norm structures of H α and H k are inherited from L 2 pGq.
In the sense of [Ver07], for a compact matrix quantum group G, we say that its discrete dual p G has the rapid decay property if there exists C, β ą 0 such that (2.10) }f } L 8 pGq ď Cp1`kq β }f } L 2 pGq for all f P H k .
Notation 2. We say that the discrete dual p G of a compact matrix quantum group G has the rapid decay property with r k À p1`kq β if the above inequality (2.10) holds.
2.4. Fourier analysis on compact quantum groups. Within the framework of compact quantum groups, the Fourier transform F : L 1 pGq Ñ ℓ 8 p p Gq, φ Þ Ñ p φ " p p φpαqq αPIrrpGq , is defined by (2.11) p φpαq i,j " φppu α j,i q˚q for all 1 ď i, j ď n α under the identification L 1 pGq " L 8 pGq˚.
If G is of Kac type, we call ÿ αPIrrpGq n α trp p φpαqu α q " ÿ αPIrrpGq nα ÿ i,j"1 n α p φpαq i,j u α j,i the Fourier series of φ P L 1 pGq and denote it by φ " ÿ αPIrrpGq n α trp p φpαqu α q. Indeed, equality f " ÿ αPIrrpGq n α trp p f pαqu α q holds for all f P PolpGq since p f pαq " 0 for all but finitely many α. It is known that the Fourier transform F : L 1 pGq Ñ ℓ 8 p p Gq is contractive, and the Plancherl theorem states that Therefore, by the complex interpolation theorem, we obtain the Hausdorff-Young inequalities ď }f } L p pGq for all f P L p pGq and 1 ď p ď 2.
2.5. Complex interpolation on vector valued ℓ p -spaces. In this section, we gather some well-known facts on complex interpolation methods extracted from [Xu96, Section 1]. For a family of Banach spaces tE k u kPZ with a positive measure µ on Z we define vector valued ℓ p -spaces by (2.14) and the natural norm structure is If pE k , F k q is a compatible pair of Banach spaces for all k P Z and µ 0 , µ 1 are two positive measures on Z, then for any 0 ă θ ă 1 we have 1 . In particular, for p 0 " 2 " p 1 and any 0 ă θ ă 1, we have Examples of compact matrix quantum groups.
2.6.1. Duals of discrete groups. Let Γ be a discrete group and Cr pΓq be the associated reduced group C˚-algebra generated by left translation operators tλ g u gPΓ . The unital˚homomorphism ∆ : Cr pGq Ñ Cr pΓq b min Cr pΓq, λ g Þ Ñ λ g b λ g , determines a compact quantum group p Γ " pCr pΓq, ∆q which we call the dual of the discrete group Γ. Then Irrp p Γq " tλ g u gPΓ and the Haar state is the vacuum state determined by h : λ g Þ Ñ δ g,e . The associated von Neumann algebra L 8 p p Γq is the group von Neumann algebra V N pΓq and L 1 p p Γq " ApΓq is called the Fourier algebra of Γ.
Moreover, if S " tg j u n j"1 Ă Γ is a generating set, then w " Cr pΓq makes p Γ into a compact matrix quantum group and Γ is said to be of polynomial growth if b k À p1`kq γ for some γ with respect to w. Moreover, if Γ has a polynomial growth, there exists a non-negative integer γ P t0u Y N such that b k « p1`kq γ . We call the non-negative integer γ the polynomial growth order of Γ.
If Γ is a non-elementary hyperbolic group, it is known that Γ p" p p Γq has the rapid decay property with r k À 1`k [Haa79,dlH88].
2.6.2. Free orthogonal quantum groups and free permutation quantum groups. Let N ě 2 and CpOǸ q be the universal unital C˚-algebra generated by tu i,j u N i,j"1 satisfying Then ∆ : pCpOǸ q, ∆q with u " pu i,j q N i,j"1 satisfies the axioms to be a compact matrix quantum group. We call OǸ free orthogonal quantum groups [Wan95].
On the other hand, we denote by CpSǸ q the universal unital C˚-algebra generated by Then ∆ : u i,j Þ Ñ N ÿ k"1 u i,k bu k,j again extends to a unital˚-homomorphism and pCpSǸ q, ∆q with u " pu i,j q N i,j"1 forms a compact matrix quantum group, which we call free permutation quantum groups SǸ [Wan98].

Standard noncommutative semigroup and examples.
Let G be a compact quantum group of Kac type. We say that a semigroup pT t q tą0 is a standard noncommutative semigroup on L 8 pGq if pT t q tą0 satisfies the following assumptions [JM12,JW17]: (1) Every T t is a normal unital completely positive maps on L 8 pGq; (2) For any t ą 0 and f, g P L 8 pGq, hpT t pf qgq " hpf T t pgqq; (3) For any f P L 8 pGq lim tÑ0`T t pf q " f in the strong operator topology.
Note that the first two conditions imply hpT t pf qq " hpf q for any f P L 8 pGq and that such a semigroup admits an infinitesimal generator L such that T t " e tL .
(1) For any connected compact Lie groups, there exists the Poisson semi- (2) For duals of non-abelian free groups, there exists the Poisson semigroup (3) For free orthogonal (resp. permutation) quantum groups OǸ (resp. SǸ`2) with N ě 2, there exists the heat semigroup pT O t q tą0 (resp. T S t ) on L 8 pOǸ q (resp. Recall that, if G is a compact matrix quantum group of Kac type whose dual has the polynomial growth b k À p1`kq γ , then for any 1 ă p ď 2 we have In this section, we will provide two sufficient conditions to prove sharpness of (3.1). One strategy requires the existence of a standard noncommutative semigroup pT t q tą0 whose infinitesimal generator L satisfies Lpu α i,j q "´lpαqu α i,j with lpαq " |α|. and the other one depends on lower bounds of the growth of k-sphere s k . These two strategies explain how we are able to obtain sharp Sobolev embedding properties for connected compact Lie groups and duals of polynomially growing discrete groups as already noted in [You18b,You18c]. Furthermore, these methods also apply to the free orthogonal quantum group O2 and the free permutation quantum group S4 .
3.1. An approach using semigroups. The purpose of this section is to extend some important techniques of [You18b, Section 6] to general compact matrix quantum groups of Kac type. Discussions in this section depend on the existence of a standard noncommutative semigroup whose infinitesimal generator behaves like the Poisson semigroup of connected compact Lie groups G.
Throughout this Section, let us suppose that there exists a standard semigroup pT t q tą0 whose infinitesimal generator L satisfies Lpu α i,j q "´lpαqu α i,j and lpαq " |α| for all α P IrrpGq. Indeed, if pP t q tą0 is the Poisson semigroup of a connected compact Lie group G, then the associated infinitesimal generator satisfies Lpu π i,j q "´κ 1 2 π u π i,j and κ 1 2 π " |π| for all π P IrrpGq.
The following is a part of [Xio17, Theorem 1.1], which is written in accordance with our notation.
Theorem 3.1 (Theorem 1.1, [Xio17]). Let G be a compact quantum group of Kac type and pT t q tą0 be a standard semigroup on L 8 pGq with the infinitesimal generator L. Then, for the semigroup pS t q tą0 " pe´tT t q tą0 and s ą 0, the following are equivalent: (1) For any 1 ď p ă q ď 8, (4) There exists 1 ă p ă q ă 8 such that › › ›p1´Lq´s Using the above Theorem, we will find the optimal s ą 0 which validates the above four equivalent statements of Theorem 3.1. Indeed, by the following Theorem 3.2, we can find out that the optimal exponent s is equal to the polynomial growth order γ under the assumption b k « p1`kq γ .
Recall that a compact quantum group G is called co-amenable if there exists a contractive approximate identity pe i q i in the convolution algebra L 1 pGq. Such a family pe i q i satisfies lim i p e i pαq " Id nα for each α P IrrpGq. It is known that every dual of a polynomially growing discrete quantum group is co-amenable [BV09, Proposition 2.1].
Theorem 3.2. Let G be a general compact quantum group of Kac type and w : IrrpGq Ñ p0, 8q be a positive function.
(1) For a given f " ÿ αPIrrpGq p α pf q P L 2 pGq, we have by the Hausdorff-Young inequality and the H: older inequality, and hence (2) Let pe i q i be a contractive approximate identity in L 1 pGq and then we may assume e i P PolpGq for all i. Then we have Hence, by taking limit as i Ñ 8, we obtain ÿ αPIrrpGq e´2 wpαq n 2 α ď C.
Corollary 3.3. Let G be a compact matrix quantum group of Kac type whose dual satisfies b k À p1`kq γ and pT t q tą0 be a standard semigroup whose infinitesimal generator L satisfies (3.9) Lpu α i,j q "´lpαqu α i,j and lpαq " |α|. 8 Then there exists a constant K " Kpsq ą 0 such that Proof.
(1) Let us assume s ě γ 2 . Then, thanks to Theorem 3.2, it is sufficient to show that , .
Since there exists a constant C ą 0 such that 1`lpαq ě Cp1`|α|q, we have Therefore, it is enough to prove that , .
From now on, let us handle the case 0 ă t ă γ 4 . Since the function f pxq " y γ e´2 y dy.
ď K 2 2 2s ă 8 Then, as in the proof of the if part, we have Since the function x Þ Ñ x γ e´x is increasing in r0, γs, we have By taking t Ñ 0`, we obtain lim tÑ0`t 2s´γ ă 8, so that 2s ě γ.
Therefore, thanks to Theorem 3.1, we obtain the following sharp Sobolev embedding property: Corollary 3.4. Let G be a compact matrix quantum group of Kac type whose dual satisfies b k « p1`kq γ and let pT t q tą0 be a standard noncommutative semigroup whose infinitesimal generator L satisfies (3.16) Lpu α i,j q "´lpαqu α i,j and lpαq " |α|. Then the following are equivalent: (1) For any 1 ă p ď 2 there exists a constant K " Kppq ą 0 such that (2) There exist 1 ă p ď 2 and a constant K ą 0 such that Example 2. For connected compact Lie groups G and the Poisson semigroup pe´t p´∆q 1 2 q tą0 , the corresponding Sobolev embedding properties are as follows: For any 1 ă p ă q ă 8 there exists a constant K " Kpp, qq ą 0 such that where n is the real dimension of G and ∆ is the Laplacian operator.

3.2.
Under the growth rate of spheres. In this section, we provide one more sufficient condition for which inequalities (3.1) are sharp without the existence of a standard noncommutative semigroup. The main ingredient is additional information on lower bounds of the growth rate of k-spheres. Indeed, for a compact matrix quantum group G, Corollary 3.6 tells us that inequalities (3.1) are sharp if its discrete dual p G satisfies (3.20) b k À p1`kq γ and s k Á p1`kq γ´1 .
Let us begin with looking at L 2 Ñ L 4 case, which is the dual of L 4 3 Ñ L 2 case. The following theorem is motivated by the proof of [You18c, Theorem 4.5.2], which is for duals of polynomially growing discrete groups.

11
Then since the sequence pwpkqq kě0 is decreasing, for any m P N. Hence, we obtain Corollary 3.6. Let G be a compact matrix quantum group of Kac type whose dual satisfies (3.24) b k À p1`kq γ and s k Á p1`kq γ´1 .
(1) For any 1 ă p ď 2 sharp Sobolev embedding properties on duals of polynomially growing discrete groups are

SOBOLEV EMBEDDING PROPERTIES UNDER THE RAPID DECAY PROPERTY
Arguably, the most important examples of compact quantum groups are duals of free groups x F N and free quantum groups such as OǸ`1, SǸ`3, Since their discrete duals are exponentially growing if N ě 2, the results from Section 3 are not applicable.
The compact quantum groups x F N , OǸ`1, SǸ`3 with N ě 2 have unique natures in view of analysis. Those are not co-amenable, the underlying C˚-algebras are non-nuclear, etc. One notable known result [You18a, Theorem 3.2] is that Hausdorff-Young inequalities can be improved in the case of free orthogonal quantum groups OǸ with N ě 3.
The aim of this Section is ‚ (Theorem 4.1) to generalize the proof of [You18a, Theorem 3.2] to general compact quantum groups whose duals have the rapid decay property and ‚ (Theorem 4.5) to combine Theorem 4.5 and [You18b, Corollary 3.9] to establish Sobolev embedding properties under the rapid decay property.

Sharpened Hausdorff-Young inequalities.
For general compact quantum groups, the Hausdorff-Young inequalities state that the Fourier transform F : L p pGq Ñ ℓ p 1 p p Gq is contractive for all 1 ď p ď 2. Moreover, boundedness of a multiplier (4.1) implies boundedness of the sequence w " pwpαqq αPIrrpGq if G is one of the compact quantum groups listed below: ‚ connected semisimple compact Lie groups ‚ duals of discrete groups Γ ‚ free orthogonal quantum group O2 ‚ quantum SU p2q group The above observation is motivated by [You18a, Section 4] and boundedness of w can be explained by [GT80, Main theorem], families of matrix elements tλ g u gPΓ , u n 0,0 ( ně0 and u n n,n ( ně0 respectively with respect to canonical choices of orthonormal bases. Nevertheless, [You18a, Theorem 3.2] has established the existence of an unbounded (exponentially) increasing sequence for the case of OǸ with N ě 3, and we will adapt the proof of [You18a, Theorem 3.2] to general compact matrix quantum groups under the rapid decay property.
Theorem 4.1 (A sharpened Hausdorff-Young inequality). Let G be a compact matrix quantum group of Kac type whose dual p G has the rapid decay property with r k À p1`kq β . Then for any 1 ă p ď 2 we have Proof. First of all, [You18b, Proposition 3.7] states that a linear map is bounded and the Plancherel identity can be interpreted as that (4.4) Φ : L 2 pGq Ñ ℓ 2´p tH k u kě0 , µq is an isometry where µpkq " p1`kq 2β . Since Xu96], the resulting inequality at θ " 2 p 1 is (4.6)˜ÿ Remark 4.2. The above Theorem 4.1 is sharp for G " x F N , OǸ`1 or SǸ`3 with N ě 2 by Corollary 6.3.
Proof. It is enough to note that Remark 4.4. In view of Corollary 4.3, if sup αPIrrpGq n α p1`|α|q 2β " 8, we are able to find an unbounded sequence w " pwpαqq αPIrrpGq such that This happens when G is one of the following: ‚ Free orthogonal quantum groups OǸ with N ě 3; ‚ Free unitary quantum groups UǸ with N ě 3; ‚ Quantum automorphism group G aut pB, ψq with a δ-trace ψ and dimpBq ě 5.

4.2.
Sobolev embedding properties under the rapid decay property. In this section, we will present Sobolev embedding properties under the rapid decay property by interpolating Theorem 4.1 and Hardy-Littlewood inequalities [You18b, Theorem 3.8].
Corollary 4.6. Let s ě 2β`1 and G be a compact matrix quantum group of Kac type whose dual has the rapid decay property with r k À p1`kq β . Suppose that there exists a standard non-commutative semigroup pT t q tą0 whose infinitesimal generator L satisfies (4.11) Lpu α i,j q "´lpαqu α i,j and lpαq " |α|. Then for any 1 ă p ă q ă 8 we have (4.12) Proof. It is enough to note that (4.13)¨ÿ Then Theorem 4.5 and Theorem 3.1 complete the proof.

RAPID DECAY DEGREE OF DISCRETE QUANTUM GROUPS
The rapid decay degree, which was suggested in [Nic10], is a way to quantify the rapid decay property of a discrete group, and the notion naturally extends to the framework of duals of compact matrix quantum groups of Kac type. A natural way is to define the degree of rapid decay property rdp p Gq as the infimum of positive numbers s ě 0 satisfying Note that this definition is independent of the choice of a generating unitary representation. For discrete groups, it has turned out that rdpΓq " γ 2 for any finitely generated discrete group Γ with the polynomial growth order γ [Nic10] and that rdpΓq " 3 2 for any nonelementary hyperbolic groups [Nic17].
As quantum analogues of the above results, we aim to compute the rapid decay degree of polynomially growing discrete quantum groups and duals of free quantum groups y OǸ , x SǸ . By theorem 3.2, we are already ready to extend [Nic10, Proposition 2.2 (2)].
Proposition 5.1. Let G be a compact matrix quantum group of Kac type whose dual satisfies b k « p1`kq γ . Then if and only if s ą γ 2 . In particular, rdp p Gq " γ 2 .
In the case that p G is exponentially growing, the following approach is valid under the rapid decay property. The proof relies on standard arguments that have been already used in [Nic10,Bra12,JPPP17,FHL`17].
Proposition 5.2. Let G be a compact matrix quantum group of Kac type whose dual has the rapid decay property with r k À p1`kq β and let w : t0u Y N Ñ p0, 8q be a positive function such that C w " ÿ kě0 p1`kq 2β e 2wpkq ă 8. Then we have In particular, for any s ą β`1 2 we have Proof. It is enough to see that for any f P PolpGq From now on, let us try to detect the rapid decay degree of duals of free quantum groups.
Theorem 5.3. Let G be a compact matrix quantum group of Kac type and w : t0u Y N Ñ p0, 8q be a positive function. If we suppose that then there exists a universal constant K ą 0 such that ÿ kě0 p1`kq 2 e 2wpkq ď KC 2 if G is one of the following: ‚ duals of non-elementary hyperbolic broups; ‚ free orthogonal quantum groups OǸ with N ě 2; ‚ free permutation quantum groups SǸ with N ě 4.
Proof. First of all, let Γ be a non-elementary hyperbolic group and σ k " ÿ gPS k λ g . Then for any positive sequence pa k q kě0 the main theorem in [Nic17] states that (5.7) pk`1qa k .
Therefore, from the given assumption, we have a k pk`1qe´w pkq .
By combining Proposition 5.2 and Theorem 5.3, we can compute the rapid decay degrees of y OǸ and x SǸ .
Proof. It is sufficient to see the only if part. By Theorem 5.3, the given assumption implies ÿ kě0 1 p1`kq 2s´2 ă 8, so that s ą 3 2 .
6. SHARP SOBOLEV EMBEDDING PROPERTIES FOR x F N , OǸ , SǸ Throughout this section, we will present sharp Sobolev embedding properties for duals of free groups and free quantum groups OǸ , SǸ . In each case, the standard noncommutative semigroups T F We begin with the following computational lemma.
Thus, it is enough to show that sup Now, we are ready to compute the optimal order of the ultracontractivity of S t " pe´tT t q tą0 .
(1) Let G " x F N with N ě 2 and T t " T F t . Then there exists a universal constant K ą 0 such that f or all f P ℓ 2 pF N q and t ą 0 if and only if s ě 3. (2) Let N ě 3, G " OǸ (resp. SǸ`2) and T t " T O t (resp. T S t ). Then there exists a universal constant K ą 0 such that Proof. Recall that, in the case of (2), T t : u k i,j Þ Ñ e´t c k u k i,j with c k " k. Now, in all cases, if we suppose s ě 3, then C w " ÿ kě0 p1`kq 2 e 2tp1`kq presp. ÿ kě0 p1`kq 2 e 2tp1`c k q q À 1 t s by Lemma 6.1 (2), so Proposition 5.2 is applicable. Conversely, from the assumed inequalities, we obtain ÿ kě0 p1`kq 2 e 2tp1`kq presp. ÿ kě0 p1`kq 2 e 2tp1`c k q q À 1 t s by Theorem 5.3, so Lemma 6.1 (2) tells us s ě 3.
Finally, since we have sharp ultracontractivity properties of S t " pe´tT t q tą0 (Corollary 6.2), we reach the following sharp Sobolev embedding properties for x F N , OǸ and SǸ by Theorem 3.1.

Example 4.
(1) Let N ě 2 and 1 ă p ă q ă 8. Then for all λpf q " ÿ gPFN f pgqλ g P L p p x F N q.

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(2) Let G " OǸ or SǸ`2 with N ě 3 and 1 ă p ď 2. Then (2) Let 1 ă p ď 2 and G be x F N , OǸ`1 or SǸ`3 with N ě 2. Then Proof. In both cases, it is sufficient to show the only if parts.