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[8, Rochberg and man Y ucinsae.Tecobtspaces coorbit The spaces. function meddcniuul n densely and continuously embedded mnsae fsvrlcomplex several of spaces gman iceesre fsmsml Lie semisimple of series discrete e h anaheeeto coorbit of achievement main The . egop,smln theory. sampling groups, ie t o ori hoyae()a (i) are theory coorbit for ata ,ψ Y ψ, π, OLAFSSON n a hndfiea define then can one ) ´ tter rvdsthe provides theory it f S ∈ Y ∗ si Co in is on n, G 1) (usually S ψ Y and if , 2 JENS GERLACH CHRISTENSEN, KARLHEINZ GROCHENIG,¨ AND GESTUR OLAFSSON´ coorbit theory attached to the holomorphic discrete series of the simple Lie groups SU(n, 1) and their covering groups. Realizing SU(n, 1)/S(U(n) U(1)) as the unit ball Bn in Cn those representations act on Hilbert spaces of holomorphic functions on B×n. The first objective is to identify and describe the coorbit space with concrete and possibly already known function spaces. The second objective then is to make the abstract atomic decompositions more concrete and formulate these explicitly. Although the step from the three-dimensional group SL(2, R) SU(1, 1) to the higher-dimensional groups SU(n, 1) is natural, the technicalities of the representation theory≃ of semisimple groups and of analysis in several complex variables are much more demanding. This may be the reason why the coorbit theory of semisimple Lie groups has mostly been untouched territory so far. To give the reader some idea of the main results, we state those without insisting in precise definitions p and formulations. For α> 1 the weighted Bergman space α is the space of all holomorphic functions on Bn such that − A p p 2 α f α = cα f(z) (1 z ) dv(z) < k k ZBn | | − | | ∞ n where dv is the Lebesgue measure on B and cα is a normalizing constant so that the constant function p 1 has norm one. The space α is a Banach space for 1 p and a Hilbert space for p = 2. We let A 2 ≤ ≤ ∞ σ = α + n + 1 and write σ = α as σ will show up in other contexts. The group SU(n, 1) consistsV A of all (n + 1) (n + 1) matrices that preserve the sesquilinear form z, w n+1 × ∈ C z1w1 znwn + zn+1wn+1. Usually, a matrix x SU(n, 1) is written in block form as →a b − −···− Bn ∈ x = ct d . As in the case n = 1, such a matrix acts on by fractional transformations. Given a real number σ>n one can then define a representation of SU(n, 1) on functions living on Bn by 1 a∗z c (1) πσ(x)f(z)= f − . ( z,b + d)σ z,b + d −h i −h i If this action is restricted to the Hilbert space σ, then πσ becomes an irreducible square-integrable unitary representation of SU(n, 1), if σ is an integer, orV one of its covering groups otherwise. Our first result identifies the coorbit spaces with respect to πσ as Bergman spaces on the unit ball. The following is an unweighted version of Theorem 3.6.
Theorem 1.1. Let 1 p < , σ > 2n max 1/p, 1 1/p , and ψ = 1Bn . The representation coefficient x f, π (x)ψ is in L≤p(SU(n,∞1)), if and only{ if f − p } . In other words, f is in the coorbit of Lp, → h σ i ∈ Aσp/2−n−1 if and only if f belongs to the Bergman space p on Bn. Aσp/2−n−1 In Theorem 3.14 we will formulate the atomic decompositions for general weighted Bergman spaces. An abbreviated version with fewer assumptions reads as follows.
Theorem 1.2. Assume that 1 p < and σ > 2n max 1/p, 1 1/p . Then there exist points wi : i I Bn, such that every f p≤ ∞possesses the decomposition{ − } { ∈ }⊆ ∈ Aσp/2−n−1 f(z)= c (f)(1 w 2)σ/2(1 z, w )−σ i − | i| − h ii Xi∈I with unconditional convergence in p . The coefficient sequence c is in ℓp and can be chosen so that Aσp/2−n−1 c is equivalent to the Bergman space norm of f. k kp We also obtain frame expansions for the Bergman spaces, which is related to the existence of sampling sequences. We are not aware of any results in this direction except in the special case of the unit disc [23, 38, 39], though they are hardly surprising considering the work by [42, 16]. The precise formulations and proofs require quite a bit of set-up. An important technical point is the integrability or lack thereof of the representations πσ for small values of σ. In fact, whereas all these representations are square integrable, for some of them there is a p′ such that the matrix coefficients are only in Lp for p>p′ > 1. In [20], and more generally [13, 24], integrability was crucial for the derivation and discretization of a reproducing formula. However, this restriction is not necessary, as is shown in [8, 9, 4, 27]. In a sense, the concrete examples of Bergmann spaces justify and support the considerable efforts spent to extend the scope of coorbit theory to non-integrable and reducible representations. It is a curious fact that Zhu’s book on Bergman spaces does not mention SU(n, 1) and its representations (though the symmetry group of the unit ball in Cn makes a guest appearance). In our approach, SU(n, 1) NEW ATOMIC DECOMPOSITONS FOR BERGMAN SPACES ON THE UNIT BALL 3 plays the main role for the understanding of Bergman spaces and their atomic decompositions. It allows us to use general methods to derive specific results for this example and guides the way for similar results for all symmetric bounded domains. This paper is organized as follows: In Section 2 we recall the definition of the Bergman spaces and the basic facts about SU(n, 1) and its discrete series. A basic lemma (Lemma 2.6) establishes the connection between the reproducing kernel of the Bergman space and the action of the discrete series of SU(n, 1) on a “distinguished” vector. In Section 3 we develop the coorbit theory for the discrete series πσ of SU(n, 1). First we recall the general definitions and concepts of coorbit theory. In Theorem 3.6 we show that the coorbits with respect to the representations πσ are precisely the weighted Bergman spaces of several complex variables on the unit ball. With this identification, the atomic decompositions of Bergman spaces and several sampling formulas can then be derived directly (though not without some effort) from the abstract results in coorbit theory.
2. Bergman spaces on the unit ball In this section we prepare the background for the complex analysis of several variables on the unit ball and for the related representation theory of Lie groups. 2.1. Bergman spaces on the unit ball. The space Cn is equipped with the inner product w,z = n n t h i w z = w1z1 + + wnzn, and the open unit ball B in C consists of n-tuples z = (z1,...,zn) for which ·2 2 ··· 2 n 2n z = z1 + + zn < 1. The ball B can be identified with the unit ball in R and thus can be equipped | | | | ··· | | 2n−1 with the measure dv = 2nr dr dσn where dσn is the rotation-invariant surface measure on the sphere 2n−1 2n 2n−1 n S in R normalized by σn(S ) = 1. Then dv is a rotation-invariant probability measure on B . 2 α Γ(n+α+1) For α > 1 define the probability measure dvα(z) = cα(1 z ) dv(z) where cα = n!Γ(α+1) , and let p n − − | | Lα(B ) be the weighted Lebesgue space carrying the norm 1/p p f p = f(z) dvα(z) . k kLα Z | | p n For α > 1 the weighted Bergman space α is the space of all holomorphic functions on B which are p n − p p A in Lα(B ). The space α is closed in Lα and hence a Banach space for 1 p < , and the space of A p n ≤ ∞ (holomorphic) polynomials is dense in α(B ). When p = 2 the space 2 is a HilbertA space with the inner product Aα
(f,g)α = f(z)g(z) dvα(z). ZBn
n γ γ1 γn For a multiindex γ = (γ1,...,γn) Z+ we write γ = γ1 + ... + γn and z = z1 zn . Then a simple calculation, see [45, Lemma 1.11], shows∈ that | | ··· γ!Γ(n + α + 1) (2) zγ 2 = . k kα Γ(n + α +1+ γ ) | | From this and the fact that the polynomials are dense in 2 one shows the following lemma: Aα Lemma 2.1. The functions Γ(n + α +1+ γ ) 1/2 ϕ (z)= | | zγ , γ Zn γ γ!Γ(n + α + 1) ∈ + form a orthonormal basis for 2 . Furthermore, if Aα 1 (3) K (z, w)= = ϕ (z)ϕ (w), α (1 z, w )n+1+α γ γ − h i Xγ then
Pαf(z)= f(w)Kα(z, w) dµα(w) ZBn is the orthogonal projection from L2 (Bn) onto 2 . In particular, for f 2 we have α Aα ∈ Aα
f(z)= f(w)Kα(z, w) dµα(w) . ZBn 4 JENS GERLACH CHRISTENSEN, KARLHEINZ GROCHENIG,¨ AND GESTUR OLAFSSON´
Thus K (z, w) is the reproducing kernel for 2 . α Aα n 2 −(n+1+α) n Note that Kα(z, w)= Kα(w,z). If w B , then Kα(z, w) (1 w ) < for all z B , and therefore the function z K (z, w) is in∈ all of the| spaces p|≤, 1 −p< | | . ∞ ∈ 7→ α Aα ≤ ∞ 2.2. Groups acting on the unit ball and integral formulas. The group SU(n, 1) consists of all matrices with determinant one that preserve the sesquilinear form J z, w = z w z w + z w , h n,1 i − 1 1 −···− n n n+1 n+1 where In 0 Jn,1 = − . 0 1 We usually write a matrix x in SU(n, 1) in block form as a b x = , ct d where a is an n n-matrix with complex entries, b,c Cn, and d C. Then by definition x is in SU(n, 1) × ∗ ∈ ∈ if and only if det x = 1 and x Jn,1x = Jn,1. This immediately gives that
∗ −1 ∗ a c (4) x = J x J = t − . n,1 n,1 b d − The group SU(n, 1) acts transitively on Bn via the fractional linear transformations a b az + b z = , ct d · c, z + d h i We denote the origin in Cn by o. Next define the subgroups K, A, and N of SU(n, 1) by k 0 K = u = k U(n) U(n) k 0 det(k) ∈ ≃
cosh(t) 0 sinh(t) A = at = 0 In−1 0 t R R ∈ ≃ sinh(t) 0 cosh(t) 2 T 2 1 z /2+ is z z /2 is − | | | | − n−1 N = nz,s = z In−1 z z C , s R − ∈ ∈ is z 2/2 zT 1+ z 2/2 is − | | | | −
From now on we let S = AN and write G = SU(n, 1). The following is well known [33, 34]: Theorem 2.2 (Two Integral Formulas). The following holds: (i) The map A N K G, (a,n,k) ank, is an analytic diffeomorphism. (ii) The Haar measures× × on→K,A,N can7→ be normalized such that for all f L1(G) ∈ f(x) dx = f(ank) dkdnda = f(hk) dkdh . ZG ZA ZN ZK ZS ZK
(iii) Both G and S act transitively on Bn. The stabilizer of o in G is K and S acts freely. Thus the mapping h h o defines equivariant maps from S and G onto Bn and yields the diffeomorphisms 7→ · Bn G/K S. ≃ ≃
(iv) The group S is simply connected and solvable. NEW ATOMIC DECOMPOSITONS FOR BERGMAN SPACES ON THE UNIT BALL 5
Right K-invariant functions on G (or, equivalently, functions on S) correspond bijectively to functions on Bn via the identification f(x)= f(x o) . · n This identification induces a SU(n, 1)-invariante measure on B , which, up to a multiplicative constant, is given by dv(z) f f(z) 2 n+1 . 7→ ZBn (1 z ) − | | We normalize this measure so that dv(z) (5) f(z) 2 n+1 = f(x) dx = f(x) dx . ZBn (1 z ) Z Z − | | G S We then obtain the following isometry between Lp-spaces.e e Lemma 2.3. Let f Lp (Bn) and let f(x)= f(x o). Then ∈ α · p p e p 2 α+n+1 p f p n := f(z) dvα(z)= cα f(x) (1 x o ) dx := cα f p Lα(B ) L (G) k k ZBn | | ZG | | − | · | k k α+n+1 and similarly for the group S. e e The integration formula (5) shows that for suitable functions the convolution on G respectively S can be written as integration over the ball. For further reference, we state this fact as a lemma. Lemma 2.4. Let f and g be functions on Bn and f and g be the corresponding functions on either G or S. Let g∨(x)= g(x−1). Then the following holds: e e (i) Assume that f is right K-invariant and g is K-biinvariant on G. Then e e dv(z) e e ∨ −1 (f g)(x)= f(z)g (x z) 2 n+1 , ∗ ZBn (1 z ) − | | whenever the convolutione one G is defined. (ii) In the case of S we have
∨ −1 dv(z) (f g)(x)= f(z)g (x z) 2 n+1 . ∗ ZBn (1 z ) e − | | 2.3. Bergman spaces and representations.e In this section we introduce the holomorphic discrete series representations of SU(n, 1) and its coverings. We also discuss the smooth vectors and the distribution vectors of those representations. This is needed to realize the Bergman spaces as coorbits. We refer to [17, 32, 44] for details and proofs. Readers with basic knowledge of representation theory can skip this subsection and use it as a reference as needed later on. The action of SU(n, 1) on Bn yields, for each σ>n, an irreducible unitary (projective) representation of SU(n, 1) on the Hilbert space = 2 by Vσ Aσ−n−1 a b 1 a∗z c (6) πσ(x)f(z)= πσ f(z)= f − . ct d ( z,b + d)σ z,b + d −h i −h i When σ is not an integer, then, because of the factor ( z,b +d)σ, π is no longer a representation of G, but −h i σ it can always be lifted to a representation of the universal covering G of G because the pullback of K in G is simply connected. If σ is rational, then there exists a finite covering G of G such that π is a well defined e σ σ e representation of Gσ. If σ is irrational, then we set Gσ = G. Then Gσ has finite center Zσ and the pullback K of K in G is compact, if and only if σ is rational. However, as the subgroup S is simply connected and σ σ e hence isomorphic to its pullback in G, the restriction πσ S of πσ to S is a well defined representation of S for all σ>n. To simplify the exposition we will carry out| most computations for S and denote π simply e σ|S by πσ. For each z Z the operator π (z) commutes with all π (x), x G . As π is irreducible and unitary ∈ σ σ σ ∈ σ σ it follows by Schur’s Lemma that there exists a homomorphism χσ : Zσ T such that πσ(z) = χσ(z) id. Hence for f,g , x G , and z Z we get → ∈Vσ ∈ σ ∈ σ (f, π (xz)g) = χ (z)(f, π (x)g) = (f, π (x)g) . | σ | | σ σ | | σ | 6 JENS GERLACH CHRISTENSEN, KARLHEINZ GROCHENIG,¨ AND GESTUR OLAFSSON´
It follows that the absolute value of the wavelet coefficient σ Wg (f)(x) = (f, πσ(x)g) is in fact a well defined function on Gσ/Zσ and hence a well defined function on SU(n, 1). Theorem 2.5. Let σ>n. Then σ (i) Let f,g σ. Then Wg (f) is square integrable on SU(n, 1). σ ∈V | | (ii) Wg (g) is integrable on SU(n, 1) for all g in a dense Gσ-invariant subspace of σ, if and only if |σ> 2n.| V The following lemma reveals the fundamental connection between the representation theory of SU(n, 1) and the sampling theory in Bergman spaces. We fix the notation ψ =1Bn , the characteristic function of the ball. a b Lemma 2.6. Let x = SU(n, 1) and f . Define w = x o = b/d. Then for ψ =1Bn ct d ∈ ∈Vσ · −σ σ −σ πσ(x)ψ(z)= d Kσ−n−1(z, w) and W ψ(f)(x)= d f(w) . Proof. The first statements follows from (3) and (6) using 1 −σ 1 −σ πσ(x)ψ(z)= = d = d Kσ−n−1(z, w) . ( z,b + d)σ (1 z,b/d )σ −h i − h i The second statement follows then immediately.
Corollary 2.7. Let ψ =1Bn , x SU(n, 1), w = x o, and f . Then ∈ · ∈Vσ (7) W σ(f)(x) = (1 w 2)σ/2 f(w) . | ψ | − | | | | In particular, for f = ψ, we have (8) W σ(ψ)(x) = (1 x o 2)σ/2 = (1 w 2)σ/2 . | ψ | − | · | − | | Proof. The explicit description of x−1 in (4) shows that d 2 b 2 = 1, and therefore | | − | | d −σ = (1 b/d 2)σ/2 = (1 w 2)σ/2 . | | − | | − | | Proposition 2.8. Let σ>n and ψ =1Bn . Then the following holds true:
(i) ψ is S-cyclic, i.e., the subspace generated by the vectors in πσ(x)ψ : x S is dense in σ. (ii) W σ(f) L2(S) for all f . { ∈ } V ψ ∈ ∈Vσ (iii) The space f W σ(f) L1(S) is non-zero if and only if σ > 2n. { ∈Vσ | ψ ∈ } Proof. (ii) and (iii) follow from Theorem 2.5 and Theorem 2.2(iii), and also from the explicit computation for Corollary 2.7. To show that ψ is S-cyclic in V , it is enough to show that W σ(f)(x) = 0 for all x S implies f = 0. σ ψ ∈ This immediately follows from (7) and the fact that S acts transitively on Bn.
2.4. Smooth vectors. The Fr´echet space of smooth vectors in σ will play an important role in the follow- ing. We therefore recall the basic definitions and facts here. LetV G be a Lie group with Lie algebra g and let π be a representation of G on a Hilbert space . A vector v is said to be smooth if the mapping x π(x)v, is smooth. The space of smooth vectorsH is denoted by∈ H ∞. It is dense in . For X g we define7→ a linear map π∞(X): ∞ ∞ by H H ∈ H → H π(exp(tX))v v π∞(X)v = lim − , t→0 t where the limit is taken in . We have H π(exp tX)π(x)v = π(x)π(exp tAd(x−1)X) . It follows therefore, by composition of smooth maps, that ∞ is G-invariant. Similarly it follows that ∞ is g-invariant. Finally, π∞ is a representation of g in ∞ suchH that for all X g and x G we have H H ∈ ∈ π(x)π∞(X)= π∞(Ad(x)X)π(x) . Recall that for SU(n, 1) the map Ad is just given by Ad(x)X = xXx−1. NEW ATOMIC DECOMPOSITONS FOR BERGMAN SPACES ON THE UNIT BALL 7
Fix a basis X ,...,X of g. For α Zk define 1 k ∈ + ∞ α α1 αk π (X) := π(X1) ...π(Xk) . We can then define a family of seminorms p on ∞ by α H (9) p (v)= π(X)αv . α k k This family of seminorms makes ∞ into a Fr´echet space such that the corresponding representation of G is continuous, in fact smooth. We denoteH the space of continuous conjugate linear functionals on ∞ by −∞. H H n The space of smooth vectors in σ can be described explicitly as a space of holomorphic functions on B with certain decay properties, whichV we now describe. This was first done for SU(1, 1) in [37] and then in general in [3, p. 551 and p. 556]. Denote by the space of homogeneous polynomials of degree k. For Pk ϕ σ write ∈V ∞ ∞ ϕ = (ϕ, ϕγ )ϕγ = (ϕ, ϕγ )ϕγ = ϕk Xγ Xk=0 |γX|=k Xk=0
where the convergence is in the norm. Thus every element in σ can be written as a sum of homogeneous polynomials. V Lemma 2.9. We have
(10) ∞ = ϕ = ϕ ϕ , and N, C, ϕ C(1 + k)−N . Vσ { k | k ∈Pk ∀ ∃ k kkVσ ≤ } Xk and (11) −∞ = ϕ = ϕ ϕ , and N,C, ϕ C(1 + k)N . Vσ { k | k ∈Pk k kkVσ ≤ } Xk Note that the dual pairing can be described as
2 σ dv(z) f, ϕ Vσ = (fk, ϕk)= fk(z)ϕk(z)(1 z ) 2 n+1 . h i ZBn − | | (1 z ) Xk Xk − | | We summarize the main properties of ∞ needed later. Vσ Lemma 2.10. For σ>n we have:
(i) Every polynomial is a smooth vector for πσ . (ii) dim ( )= k+n−1 (1 + k)n. Pk k ≤ (iii) Every smooth vector is bounded on Bn. Proof. Parts (i) and (ii) are clear. Part (iii) can be proven using results in [3], but since our situation is simpler we include a self contained proof here. We start by noting that k is a finite dimensional subspace 2 2n−1 P of L (S ) and for each z the map p p(z) is linear and well defined and hence continuous. Thus k is a reproducing kernel Hilbert space. By7→ Lemma 1.11 in [45] a polynomial of homogeneous degree k satisfiesP
2 Γ(n + k) Γ(σ) 2 Γ(n + k) Γ(σ) 2 2 2n−1 p Vσ = p(z) dσn(z)= p L (§ ) k k Γ(n) Γ(σ + k) ZS2n−1 | | Γ(n) Γ(σ + k)k k It follows from (2) that γ 2 γ! Γ(n) z 2 S2n−1 = . k kL ( ) Γ(n + k) 1/2 γ Thus (Γ(n + k)/γ!Γ(n)) z |γ|=k is a orthonormal basis for k and the reproducing kernel Hk is given by { } P
Γ(n + k) γ γ 2 n (12) H (z, w)= z w , H 2 S2n−1 S2n−1 = dim (1 + k) . k γ!Γ(n) k kkL ( × ) Pk ≤ |γX|=k As H (k z, w)= H (z, k−1w) for all k K and z, w Bn it follows from the K-invariance of the measure k · k ∈ ∈ σn and (12) that
2 2 n Hk(z, w) dσn(w)= Hk(z, w) dσn σn(z, w) (1 + k) . ZS2n−1 | | ZS2n−1×S2n−1 | | ⊗ ≤ 8 JENS GERLACH CHRISTENSEN, KARLHEINZ GROCHENIG,¨ AND GESTUR OLAFSSON´
Combining all of this, we get for p and z Bn: ∈Pk ∈ n/2 p(z) = p(w)Hk(z, w) dσn(w) (1 + k) p L2(S2n−1) . | | ZS2n−1 ≤ k k
n σ For σ>n we have Γ(n + k) /Γ(σ + k) k /k as k . Hence, there exists constants C1,C > 0 such that for all k ∼ → ∞ Γ(n + k) C kn−σ C(1 + k)n−σ . Γ(σ + k) ≥ 1 ≥ From this we see that 2 σ−n 2 p(z) dσn(z) C(1 + k) p Vσ . ZS2n−1 | | ≤ k k
Thus, if ϕ = k ϕk, with ϕk homogeneous of degree k, it follows from Lemma 10 that P ϕ(z) C (1 + k)σ/2 ϕ C (1 + k)σ/2(1 + k)−N | |≤ k kkVσ ≤ N Xk Xk which is finite if N is large enough.
3. Bergman spaces, wavelets, frames and atomic decompositions We now present the general (coorbit) theory which allows us to use the discrete series representations to provide atomic decompositions and frames for the Bergman spaces. 3.1. Wavelets and coorbit theory. In this section we describe the elements of coorbit theory with the goal of applying them to the representation πσ of S or Gσ. As πσ is highly reducible on S, we will use the general set-up of [8, 9, 4] rather than the original work in [20]. Let be a Fr´echet space and let ∗ be the conjugate linear dual equipped with the weak* topology, and assumeS that is continuously embeddedS and weakly dense in ∗. The conjugate dual pairing of elements ψ and fS ∗ will be denoted by f, ψ . Let G be a locallyS compact group with a fixed left Haar measure∈ S dx, and∈ assumeS that (π, ) is a continuoush i representation of G, i.e. x π(x)ψ is continuous for all ψ . A vector ψ is called weaklyS cyclic if f, π(x)ϕ = 0 for all x G implies7→ that f = 0 in ∗. The contragradient∈ S representation∈ S (π∗, ∗) is the continuoush representationi of∈G on ∗ defined by S S S π∗(g)f, ϕ = f, π(g−1)ϕ for f ∗. h i h i ∈ S For a fixed vector ψ define the wavelet transform W : ∗ C(G) by ∈ S ψ S → W (f)(x)= f, π(x)ψ = π∗(x−1)f, ψ . ψ h i h i A cyclic vector ψ is called an analyzing vector for if for all f ∗ the following convolution reproducing formula holds S ∈ S (13) W (f) W (ψ)= W (f). ψ ∗ ψ ψ The main achievement of coorbit theory is the discretization of the reproducing formula (13) and the construction of atomic decompositions and Banach frames for the coorbit spaces [19, 20, 30]. We formulate these results in the set-up of [4], as it also covers the case of non-integrable representations. The second ingredient of coorbit theory is a class of function spaces on G. In the sequel Y is a Banach function space on G satisfying the following properties [9]:
1 (R1) Y is a solid Banach function space continuously embedded in Lloc. Y is invariant under left and right translation of G: If ℓ f(y) = f(x−1y) denotes the left translation of a function f on G and x, y G, x ∈ ℓ f C f for all f Y for some weight function C on G. Moreover, for each f Y , the mapping k x kY ≤ xk kY ∈e e x e ∈ x ℓ f is continuous G Y . Likewise for the right translation. 7→e x e → e (R2) For a fixed analyzing vector ψ the sequilinear mapping (f,˜ ϕ) f(x) π∗(x)ψ, ϕ dx C is e ∈ S 7→ G h i ∈ continuous on Y . R × S e (R3) For a fixed analyzing vector ψ, the convolution operators f˜ f˜ Wϕ(ψ) and f˜ f˜ Wψ(ϕ) are bounded on Y for all ϕ . → ∗ | | → ∗ | | In our case, Y will be∈ a S weighted Lp-space, and the invariance properties (R1) are obviously satisfied. NEW ATOMIC DECOMPOSITONS FOR BERGMAN SPACES ON THE UNIT BALL 9
For an analyzing vector ψ define the subspace Yψ of Y by Y = f Y f = f W (ψ) , ψ { ∈ | ∗ ψ } and define the coorbit space with respect to thee wavelete ψeto be CoψY = f ∗ W (f) Y S { ∈ S | ψ ∈ } equipped with the norm f = Wψ(f) Y . k k k k ψ The following result from [9] lists conditions which ensure that CoS Y is a Banach space that is isometrically isomorphic to Yψ. Even weaker conditions can be found in [9]. Theorem 3.1. Assume ψ is an analyzing vector for and Y satisfies the axioms (R1) — (R3). Then S −1 (i) Yψ is a closed reproducing kernel subspace of Y with reproducing kernel K(x, y)= Wψ(ψ)(x y). ψ ∗ (ii) The space CoS Y is a π -invariant Banach space. ψ ∗ (iii) Wψ : CoS Y Y intertwines π and the left translation. → ∗ ψ (iv) If left translation is continuous on Y, then π acts continuously on CoS Y. (v) CoψY = π∗(f)ψ f Y . S { | ∈ ψ} (vi) W : CoψY Y is an isometric isomorphism. ψ S →e ψ e To summarize, in order to apply the abstract theory to a concrete representation π of a locally compact group, we need to verify that the representation possesses an analyzing vector ψ in a suitable space of test functions S, such the the reproducing formula (13) holds, and that the function spaces satisfies the axioms (R1) — (R3). The reproducing formula is a non-trivial property and is usually related to the square- integrability of π, see [25]. For non-integrable representions the boundedness of the convolution operators in (R2) may also be rather subtle (see Proposition 3.8). In our case the starting point is the representation (π , ) of S and of G . We take = ∞ and note σ Vσ σ S Vσ that is contained in the space of smooth vectors for πσ S and is invariant under the group S and its Lie algebra.S | To formulate sufficient density conditions for discrete sets in groups and general sequence spaces, we recall that a set X = xi : i I G is called relatively separated in G, if maxg∈G card(X gU) < for some (all) compact neighborhood{ ∈ }⊆U G. Given a neighborhood U G, X is called U-dense,∩ if ∞x U = G. ⊆ ⊆ i∈I i The following theorem is a general version of the main theorem of coorbit theory and yieldsS the existence of atomic decompositions and frames for coorbit spaces. Theorem 3.2. Let G be a connected Lie group and (π, ) a unitary representation of G. Assume that is a Fr´echet space continuously embedded in , and that His invariant under G and its Lie algebra. Let ψ S be an analyzing vector and Y be a BanachH space of functionsS on G satisfying the conditions (R1) — (R3).∈ S
(A) Atomic decompositions for coorbit spaces. There exists a neighborhood U G, such that every relatively separated and U-dense set X = x i I generates an atomic decomposition⊆ π(x )ψ i I { i | ∈ } { i | ∈ } for CoψY . This means that there exist constants A,B > 0 and a set of functionals c i I (CoψY )∗, S { i | ∈ }⊆ π such that every f CoψY possesses the expansion ∈ S
(14) f = ci(f)π(xi)ψ . Xi∈I ψ The series (14) converges unconditionally in CoS Y , whenever the functions with compact support are dense in Y , otherwise the series is weak∗-convergent in S∗. The coefficient sequence satisfies
ci(f)1x U Y A f ψ . k i k ≤ k kCoS Y Xi∈I ψ Conversely, if ai1x U Y , then h = aiπ(xi)ψ is in Co Y and h ψ B ai1x U Y . i∈I i ∈ i∈I S k kCoS Y ≤ k i∈I i k P P P (B) Banach frames for coorbit spaces. There exists a neighborhood U G, such that every relatively ⊆ separated and U-dense set X = x i I generates a Banach frame π(x )ψ i I for CoψY . This { i | ∈ } { i | ∈ } S 10 JENS GERLACH CHRISTENSEN, KARLHEINZ GROCHENIG,¨ AND GESTUR OLAFSSON´ means that there exist constants A,B > 0 and a dual frame e i I , such that, for all f CoψY { i | ∈ } ∈ S
A f ψ f, π(xi)ψ 1x U Y B f ψ , k kCoS Y ≤k h i i k ≤ k kCoS Y Xi∈I and f = f, π(x )ψ e h i i i Xi∈I
ψ ∗ with unconditional convergence in CoS Y , when the functions with compact support are dense in Y and weak in S∗ otherwise.
Proof. (A) Under the stated conditions the reproducing kernel space Yψ with reproducing kernel Φ = Wψ(ψ) satisfies both Theorem 2.13 and 2.14 from [4]. Note that by unitarity left and right derivatives of the kernel ∞ (refer to [4]) correspond respectively to the functions Wψ(π (X)ψ) and Wπ∞(X)ψ(ψ)(x) for X g. Since is π∞-invariant we therefore see that convolution with left or right derivatives of Φ will be continuous∈ on YS. This is enough to ensure the invertibility of the appropriate operators in [4] leading to atomic decompositions. (B) Since right differentiation of the kernel Φ = Wψ(ψ) corresponds to the function Wπ∞(X)ψ(ψ) and is π∞-invariant, the conditions ensure that the required convolution in Lemma 2.8 from [4] is continuous.S Therefore Theorem 2.11 from [4] provides the desired frames.
p Remark 3.3. p If Y is a weighted Lebesgue space, Y = Lv(G), then i∈I ci1xiU Lv is equivalent to the 1/p k k p p p P weighted ℓ -norm c p = c v(x ) . Thus the implicit definition of the sequence space norm is k kℓv i∈I | i| i perfectly natural. P
3.2. Coorbits for the discrete series representation. In the following sections we apply the abstract theory to the representation πσ on either Gσ or S and will show that the weighted Bergman spaces arise as coorbits for the groups S and Gσ. Then we derive atomic decompositions and frames for these spaces. We will mainly use the set-up in [4]. As mentioned earlier we take for the space of smooth vectors in σ. For ∞ S V the moment we also fix the notation ψ =1Bn σ . We start by a simple observation. Let x G∈Vand set w = x 0. Let f −∞ and write f = f as in ∈ σ · ∈Vσ k k (11) and note that this sum converges in the weak topolgy. Hence we may interchange the sumP with f and obtain 1 1 (15) W σ(f)(x)= W σ(f )(x)= f (w)= f(w), ψ ψ k dσ k dσ Xk Xk since each f is in . k Vσ ∞ Proposition 3.4. Let ψ =1Bn and let σ>n, then ψ is both S-cyclic and Gσ-cyclic in σ . If ϕ is a smooth σ σ V vector, then there is a constant C such that Wψ (ϕ) C Wψ (ψ) . Furthermore, for 1 p< the function σ p | |≤ | | σ p ≤ ∞ Wψ (ϕ) is in Lt (S) for t + pσ/2 > n. If σ is rational, then also Wψ (ϕ) Lt (Gσ) for the same range of parameters. ∈
Proof. Cyclicity for both S and Gσ follows from equation (15). By Lemma 2.6 and the fact that ϕ σ we obtain that W σ(ϕ)(x)= 1 ϕ(x o)= W σ(ψ)(x)ϕ(x o). Since ϕ is bounded on Bn by Lemma 2.10,∈ the V ψ dσ · ψ · estimate follows. By Corollary 2.7 we have
W σ(ψ)(x) = (1 x o 2)σ/2 , | ψ | − | · | so W σ(ψ) is in Lp(S) if and only if the following integral is finite | ψ | t
(16) (1 x o 2)pσ/2+t dx = (1 z 2)t+pσ/2−n−1 dv(z), ZS − | · | ZBn − | | which happens when t + pσ/2 >n. If σ>n is rational, then the integral over Gσ reduces to a finite multiple of the integral over S, and the proof is complete. NEW ATOMIC DECOMPOSITONS FOR BERGMAN SPACES ON THE UNIT BALL 11
3.3. Bergman spaces as coorbits for the group S. In this section we verify the assumptions of The- orem 3.1 for the representations πσ on S and then identify the weighted Bergman spaces as coorbit spaces with respect to πσ S. For simplicity we write πσ for πσ S as it will always be clear if we are using S or Gσ. | p | First we show that the coorbit spaces CoLα+n+1−σp/2(S) with respect to the representation πσ are well defined. In this case we deal with a weighted Lp space on S with weight (1 x o 2)t for some t. This space is invariant under left and right translations because the weight is easily seen− | · to| be moderate (a property which is equivalent to translation invariance).
Proposition 3.5. Let ψ =1Bn , σ>n and 1 p< . (i) Then there is a positive constant C such≤ that ∞ W σ(f) W σ(ψ)= CW σ(f) for every f −∞ , ψ ∗ ψ ψ ∈Vσ where convolution is on the group S. Thus √Cψ is an analyzing vector for πσ. (ii) If 1 <α For the continuity it is enough to show that f f(x)W σ (ψ)(x) dx is continuous. This will follow 7→ | ψ | provided that W σ(ψ) is in the K¨othe dual of Lp R (S) which is Lq (S) where 1/p+1/q = ψ α+n+1e −σp/2 e σq/2−(α+n+1)q/p 1. According to Proposition 3.4 this is the case if σq/2 (α + n + 1)q/p + σq/2 >n which is equivalent to α p Theorem 3.6. Let α > 1, 1 p < , and ψ = 1Bn . A holomorphic function f is in α if and only if −∞ σ − p ≤ ∞ p A f σ and Wψ (f) is in Lα+n+1−σp/2(S). Therefore the Bergman space α is equal to the coorbit space ∈ψ V p A Co ∞ L (S) for 1 <α Since zγ 2 = zγ 2 = γ! Γ(σ) by (2), we obtain that k kVσ k kσ−n−1 Γ(σ+|γ|) ∂γf 2 f 2 dim( ) (0) zγ 2 k kkVσ ≤ Pk ∂zγ k kVσ |γX|=k Γ(σ ) ∂γf 2 1 = dim( ) (0) . Pk Γ(σ + k) ∂zγ γ! |γX|=k Next we differentiate the reproducing formula for 2 in Lemma 2.1 and note that we can interchange the Aβ integration and differentiation to get γ ∂ f Γ(β + n +1+ k) γ γ (0) = f(w)w dvβ (w) . ∂z Γ(β + n + 1) ZBn 1/2 Γ(β+n+1+k) γ 2 Using the orthonormal basis ϕγ (z)= γ!Γ(β+n+1) z for β, we see that A 2 Γ(β + n +1+ k) γ 2 f(w)w dvβ (w) = f, ϕγ A2 γ! Γ(β + n + 1) ZBn |h i β | and consequently 2 Γ(β + n +1+ k) Γ(σ) 2 fk V dim( k) f, ϕγ A2 k k σ ≤ P Γ(β + n + 1) Γ(σ + k) |h i β | |γX|=k Γ(β + n +1+ k) Γ(σ) 2 dim( k) f A2 ≤ P Γ(β + n + 1) Γ(σ + k) k k β Since clearly the sequence Γ(β + n +1+ k) Γ(σ) c = k Γ(β + n + 1) Γ(σ + k) is bounded in k and dim( ) (1 + k)n, it follows that Pk ≤ 2 n fk C(1 + k) , k kVσ ≤ and therefore f = f −∞ by the definition of −∞. k k ∈Vσ Vσ P p Note that the Bergman space α can be represented as a coorbit space with respect to all representa- α+1 A tions πσ for which σ>n + p . This independence of the representation is an interesting and surprising phenomenon and is perhaps related to the semisimplicity of Gσ. 3.4. Atomic decompositions of Bergman spaces via the group S. In this section we construct frames and atomic decompositions for the Bergman spaces by applying the abstract Theorem 3.2. We first need to verify the continuity of the convolution operator f f W σ(ψ) . For that we recall the → ∗ | ψ | following well known fact from [45, Thm. 2.10]: Theorem 3.7. Fix two real parameters a and b and define the integral operator S by (1 w 2)b S 2 a f(z)=(1 z ) − | n|+1+a+b f(w) dv(w). − | | ZBn 1 z, w | − h i| Then, for t real and 1 p< , S is bounded on Lp(Bn) if and only if pa Proposition 3.8. If 1 p< and 1 <α σ σ −1 (f Wψ (ψ) )(x)= f(y) Wψ (ψ)(y x) dy ∗ | | ZS | | e e 1 = f(y) σ dy Z d −1 S | y x| e 2 σ/2 2 σ/2 (1 z ) dv(z) = (1 w ) f(z) − | | σ 2 n+1 . − | | ZBn 1 w,z (1 z ) | − h i| − | | This convolution operator corresponds to S in Theorem 3.7 with a = σ/2, b = σ/2 n 1. Applying p n p − − Theorem 3.7, we see that this operator is bounded on Lα−σp/2(B ) and thus on Lα+n+1−σp/2(S), if and only if pσ/2 < α σp/2+1 Theorem 3.11. Let 1 p< and assume that 1 <α The proof shows a bit more than just the existence of a sampling set X. In fact, for ǫ> 0 small enough every relatively separated and Uǫ-dense set X S generates a Banach frame πσ(xi)ψ : i I . Since by 2 σ/2 ⊆ { p ∈ } Corollary 2.7 f, πσ(x)ψ = (1 w ) f(w), the existence of Banach frames for α is in fact a sampling theorem in disguise.h i − | | A Corollary 3.13. Assume that 1 <α p 2 α+n+1−σp/2 p (18) A f p c (f) (1 w ) B f p for all f . k kAα ≤ | i | − | i| ≤ k kAα ∈ Aα Xi∈I Proof. This is just Theorem 3.2(A) for the concrete representation π on S and the identification of p with σ Aα the coorbit space Coψ Lp (S). Consequently every f p possesses an atomic decomposition of πσ α+n+1−σp/2 ∈ Aα the form f = ci(f)πσ (xi)ψ Xi∈I 2 σ/2 with a coefficient sequence depending linearly on f and satisfying (18). Since πσ(xi)ψ = (1 wi ) (1 z, w )−σ, this is the theorem, as stated. − | | − h ii Again, coorbit theory asserts that every Uǫ-dense and separated subset of S (for ǫ small enough) generates an atomic decomposition (with w = x o). i i · We would also like to remark that in principle the coefficients ci(f) and the dual Banach frame can be calculated by inverting a frame-like operator. See [19, 30] for details. 3.5. Bergman spaces as coorbits for the group Gσ when σ is rational. So far we have worked with the group S instead of the full group Gσ. The advantage of using S is that there is no restriction on the parameter σ (except square-integrability forcing σ>n). On the other hand, the representation πσ S is reducible on 2 , and thus the derivation of the reproducing formula is non-trivial and was done| only Aσ−n−1 for the special analytic vector ψ =1Bn . In this section we study the Bergman spaces as coorbit spaces with respect to the full group Gσ. In this case, we have to impose the restrictions σ>n and σ being rational so that πσ is square-integrable. Since πσ is irreducible and square integrable, the orthogonality relations ([15, Thm. 14.3.3] or [25, 20]) imply that the reproducing formula (13) holds automatically for all (appropriately normalized) vectors ζ σ. We then gain considerable flexibility and generality in the choice of the expanding vector. ∈V The technical work for the transition from S to Gσ is minimal and consists of modifications of the arguments in the previous sections. NEW ATOMIC DECOMPOSITONS FOR BERGMAN SPACES ON THE UNIT BALL 15 Lemma 3.15. If ζ and ϕ are smooth vectors for πσ, there is a constant Cϕ,ζ depending on ϕ and ζ such that W σ(ϕ)(x) C W σ(ψ)(x) 1 log(1 x o 2) = C (1 x o 2)σ/2 1 log(1 x o 2) | ζ |≤ ϕ,ζ | ψ | − − | · | ϕ,ζ − | · | − − | · | a b Proof. We recall the notation x = . By Lemma 2.10(iii) the functions ζ and ϕ are bounded on Bn, ct d so C = sup Bn ϕ(w)ζ(z) is finite. Furthermore, ϕ,ζ w,z∈ | | −σ −1 −σ −σ πσ(x)ζ(z) = d¯ z,b ϕ(x z) d 1 z,b/d sup ϕ(z) , | | | − h i| | · | ≤ | | | − h i| z∈Bn | | therefore σ 2 σ−n−1 Wζ (ϕ)(x) = ϕ(z)πσ(x)ζ(z)(1 z ) dz | | ZBn − | | 1 1 2 σ−n−1 Cϕ,ζ σ σ (1 z ) dz ≤ d ZBn 1 z, x o − | | | | | − h · i| According to Theorem 1.12(2) in [45] this integral is comparable to log 1 as x o 1−. Since 1−|x·o|2 | · | → W (ψ)(x)= d−σ = (1 x o 2)σ/2, the desired estimate follows. ψ − | · | Recall, that if σ is rational, then Gσ is a finite covering of SU(n, 1). Hence the inverse image of K is compact and the center of Gσ is finite. Proposition 3.16. Let α> 1 and 1 p< be given and choose a rational σ>n such that 1 <α< p(σ n) 1. Assume that ζ and− ϕ are≤ smooth∞ vectors for π . Then − − − σ (i) ζ is G -cyclic in ∞ and W σ(ϕ) Lp(G ) for t + pσ/2 >n. σ Vσ ζ ∈ t σ (ii) For f −∞ there is a constant C such that ∈Vσ W σ(f) W σ(ζ)= CW σ(f) ζ ∗ ζ ζ 2 with convolution on Gσ. In fact, C = ∆σ ζ A2 where ∆σ is the formal dimension of πσ. k k σ−n−1 (iii) The mapping p ∞ ∗ Lα+n+1−σp/2(Gσ ) σ (f, ϕ) f(x) π (x)ζ, ϕ dx C ×V ∋ 7→ ZGσ h i ∈ is continuous. (iv) f p if and only if f −∞ and W σ(f) Lp (G ). ∈ Aα ∈Vσ ζ ∈ α+n+1−σp/2 σ p ζ p (v) Up to normalization of ζ, α is equal to the coorbit CoV∞ L (Gσ). A σ α+n+1−σp/2 Proof. (i) Since πσ is irreducible, every non-zero vector is cyclic. Next we prove that W σ(ϕ) Lp(G ) for t + pσ/2 >n. Since x (1 x o 2)ǫ log(1 x o 2) is bounded ζ ∈ t σ → − | · | − | · | on G for every ǫ> 0, we may choose ǫ and C > 0, such that t + p(σ ǫ)/2 >n and (by Lemma 3.15) σ − W σ(ϕ)(x) C (1 x o 2)σ/2 1 log(1 x o 2) C(1 x o 2)(σ−ǫ)/2 . | ζ |≤ ϕ,ζ − | · | − − | · | ≤ − | · | The weight x (1 x o 2) is K-right-invariant on G , therefore the decomposition of the Haar measure 7→ − | · | σ on Gσ described in Theorem 2.2 yields σ p 2 t Wζ (ϕ)(x) (1 x o ) dx ZGσ | | − | · | − 2 (σ ǫ)p 2 t C (1 x o ) 2 (1 x o ) dx ≤ ZGσ − | · | − | · | − 2 (σ ǫ)p +t = C (1 sk o ) 2 dk ds ZS ZKσ − | · | − 2 σ ǫ p+t = C (1 s o ) 2 ds , ZS − | · | and the last integral is finite by Proposition 3.4. Thus W σ(ϕ) Lp(G ) for t + pσ/2 >n. ζ ∈ t σ 16 JENS GERLACH CHRISTENSEN, KARLHEINZ GROCHENIG,¨ AND GESTUR OLAFSSON´ (ii) The reproducing formula follows from the orthogonality relations for representation coefficients. As −∞ shown in Proposition 3.2 in [8], the reproducing formula extends from the Hilbert space to f σ . σ q ∈V (iii) As in the proof of Proposition 3.8 it suffices to show that Wζ (ϕ) is in Lσq/2−(α+n+1)q/p(Gσ) where 1/p +1/q = 1. By item (i) this is the case, if σq (α+n+1)q + σq > n, which amounts exactly to our 2 − p 2 assumption that α η = πσ (x)ϕ, ψ πσ (x)ζ dx = ϕ, ζ ψ ZGσ h i h i is also smooth (just a multiple of ψ), and therefore the mapping f f, πσ(x)ζ πσ (x)ϕ, ψ dx = f, η 7→ ZGσ h ih i h i −∞ is weakly continuous from σ to C. V p By Corollary 3.10 the convolutions by both Wψ(ζ) and Wζ (ψ) on Lα+n+1−σp/2(Gσ) are continuous, and thus the conditions of [9, Theorem 2.7] have been verified. 3.6. Atomic decomposition via the group Gσ. We will now show that every smooth vector, and in particular every polynomial, provides atomic decompositions and frames for the Bergman space. To our knowledge this is a new result in the analysis of several complex variables and goes beyond the atomic decompositions of Coifman and Rochberg [11]. In dimension 1 this consequence of coorbit theory was observed in [38]. Theorem 3.17. Assume that σ>n is rational, 1 p < , and 1 <α Proof. Again we apply Theorem 3.2 to the representation πσ of the group Gσ (instead of S) and the vec- tor ζ (instead of ψ). In Proposition 3.16 we have already verified the assumptions that guarantee that ζ p Co ∞ L (Gσ) is well defined. Vσ α+n+1−σp/2 For the atomic decompositions and Banach frames we only need to verify the additional property that the convolution operator f f W σ(ϕ) is continuous on Lp (G ) for every smooth vector ζ 7→ ∗ | ζ | α+n+1−σp/2 σ and ϕ. Since W σ(ϕ)(x) C W σ(ψ)(x) 1 log(1 x o 2) and the latter function is K-bi-invariant, the | ζ |≤ | ψ | − − | · | convolution operator reduces to an integral over the group S and by Lemma 2.4 to an integral over Bn. We may therefore assume that f is in Lp (S) and corresponds to a function f Lp (Bn). The α+n+1−σp/2 ∈ α−σp/2 e NEW ATOMIC DECOMPOSITONS FOR BERGMAN SPACES ON THE UNIT BALL 17 only difference from the calculation on S in (17) is the factor log(1 x o 2) arising from Lemma 3.15. − − | · | Since W σ(ψ)(x) = (1 x o 2)σ/2, we obtain for every ǫ> 0 (to be chosen later) | ψ | − | · | f(x) W σ (ψ)(x−1y) 1 log(1 x−1y o 2) dx Z | || ψ | − − | · | S e C f(x) (1 x−1y o 2)σ/2−ǫ dx ≤ ZS | | − | · | e 2 (σ−ǫ)/2−n−1 2 (σ−ǫ)/2 (1 w ) = C(1 z ) f(w) − | | σ−ǫ dv(w) − | | ZBn | | 1 z, w | − h i| This corresponds to the operator S in Theorem 3.7 with a = (σ ǫ)/2 and b = (σ ǫ)/2 n 1. According to Theorem 3.7 the operator S is continuous on Lp (Bn) if− and only if (σ −ǫ)p/2−< α− σp/2+1 < α−σp/2 − − − p[(σ ǫ)/2 n]. Since the assumption 1 <α Remark 3.18. We would like to point out that the group N is isomorphic to the Heisenberg group and that the group A acts by dilations. Hence one can use the coorbit theory to study Besov spaces and their atomic decompositions on the unit ball [45, Chapter 6] using the theory in [7, 28]. 3.7. Further Remarks. In the case of integrable representations, i.e., σ > 2n and σ Q, the original coorbit theory in [19, 20, 30] can be applied directly and many of the technical details∈ can be omitted. For instance, if W σ(ψ) L1(G ), then the convolution operator f f W σ(ψ) is bounded on Lp(G ) ζ ∈ σ → ∗ | ζ | σ and the subtleties of Theorem 3.7 are not needed. In this case the set of atoms used in Theorem 3.17 can be significantly enlarged. For simplicity, we discuss this effect only for coorbits of the unweighted Lp- ζ p p ζ p spaces on Gσ, which are Co L (Gσ) = . If σ > 2n, σ Q, then the definition of Co L (Gσ ) Vσ Aσp/2−n−1 ∈ Vσ ζ 1 1 ∞ is independent of ζ Co L (Gσ) = rather than just ζ [20, Thm. 4.2]. Likewise, in ∈ Vσ Aσ/2−n−1 ∈ Vσ Theorem 3.17 the atom ζ for the atomic decompositions and frames can be chosen in a space that is strictly between ∞ and 1 (denoted by in [20, Thm. 4.2]). Vσ Aσ/2−n−1 Bw As a further benefit of the use of integrable representations we mention the construction of Riesz sequences in certain Bergman spaces. Recall that for a given compact set K G a set X = x is called K- ⊆ σ { i}i∈I separated, if the translates xiK are pairwise disjoint. ∞ Proposition 3.19. Assume that σ > 2n, σ Q, 1 (19) A c ℓp(X) ciπσ(xi)ζ Ap B c ℓp(I) k k ≤k k σp/2−n−1 ≤ k k Xi∈I for all c ℓp(X). ∈ σ q ζ q Proof. By [21, Thm. 8.2] the wavelet transform W is interpolating on = Co L (Gσ) in the sense ζ Aqσ/2−n−1 Vσ that there exist a compact set K Gσ and a constant C > 0 with the following property: If X = xi i∈I is K-separated and d ℓq(X), then⊆ there exists a function f q , such that { } ∈ ∈ Aqσ/2−n−1 σ (f, πσ (xi)= Wζ (f)(xi)= di i and f Aq C d q . ∀ k k qσ/2−n−1 ≤ k k p q Now let c ℓ (X) and q the conjugate index. By duality choose d ℓ (X) with d q = 1 and i∈I dic¯i = ∈ q ∈ k k σ c p. According to the interpolation property there is an f qσ/2−n−1 such that (f, πσ(xi)ζ)= WPζ (f)(xi)= k k q ∈ Aζ p q q di and f A C d q = C. Since = CoV ∞ L (Gσ) is the dual space of = k k qσ/2−n−1 ≤ k k Aσq/2−n−1 σ Aσp/2−n−1 18 JENS GERLACH CHRISTENSEN, KARLHEINZ GROCHENIG,¨ AND GESTUR OLAFSSON´ ζ p CoV ∞ L (Gσ) for 1 f Aq ciπσ(xi)ζ Ap ≤k k σq/2−n−1 k k σp/2−n−1 Xi∈I C ciπσ(xi)ζ Ap . ≤ k k σp/2−n−1 Xi∈I The upper bound in (19) always holds by Theorem 3.2. 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