Hardy-Littlewood inequalities on compact quantum groups of Kac type

0.29 MB·

by  SangGyun Youn

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 1<p 2, there exists a constant C >0 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 inequal-
ities 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 inequali-
ties 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 < p <
1
≤
p < . Assume that a sub-linear map A : M L1(N) satisfies the following:
2
∞ →
There exists C1,C2 > 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 <p < . Assume that A: L (G) c(Irr(G),ν) is sub-linear and satis≤fies the
2 ∞
following:∞There exists C1,C2 >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 in-
terpolation 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 
1<p 2, there exists a universal constant K =K(p)>0 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 1<p 2.
≤
First,
(Af)(α) 2 ν(α)= n f(α) 2 = f 2 .
|| ||HS α|| ||HS || ||L2(G)
α∈XIrr(G) α∈XIrr(G)
b
This implies that A is of (strong) type (2,2) with C =1.
2
6
f(α) tr(f(α) f(α))
HS ∗ 1
Second,forally >0,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 sub-
section, 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 1<p 2, there exists a universal constant K =K(p)>0 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 1<p 2.
{ }∪ ≤
Firstly,
(||pk(f)||L2(G))2ν(k)= p (f) 2 = f 2 .
w(k) || k ||L2(G) || ||L2(G)
k 0 k 0
X≥ X≥
Therefore, A is of (weak) type (2,2) with C =1.
2
Secondly, for all y >0,
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 < p 2, there exists a universal constant K = K(p) > 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