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New atomic decompositions of Bergman spaces on bounded symmetric domains

We provide a large family of atoms for Bergman spaces on irreducible bounded symmetric domains. This vastly generalizes results by Coifman and Rochberg from 1980. The atomic decompositions are derived using the holomorphic discrete series representations for the domain, and the approach is inspired by recent advances in wavelet and coorbit theory. This approach also settles the relation between atomic decompositions for the bounded and unbounded realizations of the domain.


Introduction
This paper is concerned with providing atomic decompositions of Bergman spaces on bounded symmetric domains. The results extend similar results by Coifman and Rochberg [14] carried out for Bergman spaces on the unbounded realization of the domain and on the unit ball. Coifman and Rochberg asked if their decompositions would hold for the bounded domains, and in this paper we give a positive answer to this question. Moreover, we rectify an issue occuring in higher rank spaces which was pointed out in a remark on p. 614 in [2] (see also Remark 4 in [16]). While an extension to the bouned domains was predicted by Faraut and Koranyi in the introduction of [16] our results provide a much larger class of atoms than have previously been discovered. The usual atomic decompositions of Bergman spaces arise from a discretization of the integral reproducing formula, and atoms can thus be regarded as samples of the Bergman kernel in one of the variables. It turns out that this result can be formulated in terms of the holomorphic discrete series representation, in which case the classical atoms correspond to letting a discrete subset of the group of isometries act on a constant function. This viewpoint is extended widely in this paper where we show that atoms can be obtained by translates of any polynomial, or more generally, any analytic function with rapidly decreasing coefficients. Similar results have been obtained by Pap [19] on the unit disc (using the Blaschke group) and by the authors and their collaborators [10,9] in the case of the unit ball.
In the last section of the article we use representation theory to explain the relation between atomic decompositions of Bergman spaces in the bounded and the unbounded realization. This issue was unresolved in [14] and therefore their Theorem 2 did not transfer directly to bounded domains for all parameters (see beginning of §2 on p. 14 in [14]). Representation theory lifts the construction of atomic decompositions from discretizing a reproducing formula on the domain to discretizing a convolution reproducing formula of matrix coefficients for the group of isometries. In particular, we use that the matrix coefficients have the same decay for the two realizations of the domain.
The motivation for our approach comes from coorbit theory for invariant Banach spaces of functions and distributions which was initiated by Feichtinger and Gröchenig [17,18] and subsequently generalized in [13,11,7,12,10,9] in order to treat a wider class of (projective) representations. In this paper we have chosen to state most results without referring to coorbit theory. The main reason for doing so is that some results can be stated with fewer restrictions than if coorbit theory were used, and also the paper would be a lot longer if we had to introduce the entire coorbit machinery.

Main results
In this section we state the main results of this article. The following sections will then be devoted to proving those statements. We also introduce the minimal set of notation that will be needed to formulate the statements.
Let D be a bounded symmetric domain in C n containing the origin o. Let G be a connected Lie group locally isomorphic with the group of isometries of D. Then G acts transitively on If G is linear as we will always assume, or more generally, with finite center, then K is compact.
If nothing else is stated, let dz denote the normalized Euclidean measure on D and define A 2 (D) to be the Hilbert space of holomorphic functions f : D → C for which is a reproducing kernel Hilbert space. Denote the reproducing kernel (Bergman kernel) by K(z, w) so that for f ∈ A 2 (D). The map K(z, w) is holomorphic in the first variable and antiholomorphic in the second variable. Furthermore K(w, z) = K(z, w).
Let J(x, z) be the complex Jacobian of the G-action on D at the point z. Then z → J(x, z) is holomorphic, and the chain rule implies that (2.2) J(xy, z) = J(x, y · z)J(y, z).
In particular, since J(e, z) = 1 as e acts trivally, it follows that Furthermore it follows that if f ∈ A 2 (D) then J(x, z)f (x · z) is also in A 2 (D) which implies by simple calculation using the reproducing property that, see Lemma 3.1, The genus of D is the number g := (n + n 1 )/r where n = dim C D, n 1 is the complex dimension of the maximal complex subdomain of D of tube type and r is the rank of D, the dimension of a maximal totally geodesic euclidean submanifold. Define the kernel h(z, w) = K(z, w) −1/g , and by abuse of notation let h(z) = h(z, z). For x, y ∈ G also let h(x, y) = h(x · 0, y · u) and h(x) = h(x, x). Note that by definition h(x, y) is right K-invariant.
The function h(z, z) γ−g = K(z, z) 1−γ/g is integrable if and only if γ > g−1. For γ > g−1 let c −1 γ = D h(z, z) γ−g dz and dµ γ (z) = c γ h(z, z) γ−g dz. Then µ γ is a quasi-invariant probability measure on D. Let L p γ (D) be the space of equivalence classes of measurable functions on D for which Define the weighted Bergman space A p γ (D) to be the subspace of holomorphic functions in L p γ (D), i.e. A p γ (D) = L p γ (D)∩O(D) with norm inherited from L p γ (D). We note that the space of holomorphic polynomials P [C n ] is dense in A p γ (D). The space, A p γ (D) is a reproducing kernel Banach space. Denote the reproducing kernel by K γ (z, w), which then satisfies . See for example Theorem 3 in [22]. We have is not necessarily defined globally on the group G, but it is always defined on the universal covering group G of G. The function j γ satisfies the cocycle relation j γ (xy, z) = j γ (x, y · z)j γ (y, z). Furthermore defines a representation of G on A 2 γ (or a projective representation of G) and (2.1) implies that the representation is unitary. It is well known that it is also irreducible.
For a holomorphic function f on D decompose it into homogeneous polynomials f k of degree k, i.e.
Define the holomorphic functions with rapidly decreasing coefficients S γ by and the space of holomorphic functions with moderately growing coefficients S * γ by The space S γ is invariant under π γ , and π γ is a projective representation of G on S γ . The space S * γ is the dual of S γ and the dual pairing of f ∈ S * γ and g ∈ S γ is given by For ψ ∈ S γ define the wavelet transform W γ ψ : Notice that the function |W γ ψ (f )| defines a continuous function on G, so we will often allow ourselves to write |W γ ψ (f )(x)| for x ∈ G. Also, the wavelet transform is injective if ψ is nonzero, due to the fact that the representation π γ restricted to S γ is irreducible. This follows since S γ are the smooth vectors for the representation π γ by [6], and by [5,Proposition 2.6] π γ restricted to the smooth vectors is irreducible if and only if π γ is irreducible on A 2 γ (D). Define L p α (G) as the space of equivalence classes of functions on the group G for which where dµ G (x) = dx denotes the suitably normalized left-invariant Haar measure on G.
We are now ready to present the first result of this paper, which gives a wavelet characterization of the Bergman spaces.
Moreover, the norms f A p α and W γ ψ (f ) L p α−γp/2 (G) are equivalent. Remark 2.2. (1) If the rank of the space is large, then γ has to be chosen sufficiently large in order to have the second part of the wavelet description.
(2) We conjecture that it is possible to avoid this restriction and to show that the Bergman spaces A p α have a wavelet characterization for the entire range g −1 < α < p(γ −g +1)+g −1 for general ψ ∈ S γ .
(3) Due to the coorbit theory for projective representations in [9] the collection } is a non-trivial Banach space for the entire range of g − 1 < α < g − 1 + p(γ − g + 1) when ψ is a polynomial. By Theorem 2.1 these spaces are Bergman spaces for α in the smaller interval from Theorem 2.1, and we expect that this is also true in the larger interval.
For a fixed countable collection of points {x i } i∈I in G define the weighted sequence space ℓ p α (I) as the sequences {λ i } for which the norm is finite. We are now ready to state the main result of this paper which provides atomic decompositions for Bergman spaces.
there is a countable discrete collection of points {x i } i∈I in G and associated functionals Notice that the range of parameters in Theorem 2.3 is smaller than the range of parameters for which atomic decompositions have been found for tube type domains. See [8] in which atomic decompositions are constructed for tube type domains T = V + iΩ by taking Laplace extensions of Besov spaces of distributions supported on the cone Ω. These atoms do not arise as samples of the Bergman kernel. Also see the Arxiv preprint [3] where atoms are obtained for the same spaces but with atoms determined by the Bergman kernel.
Remark 2.5. In the case of ψ = 1 it is possible to choose the points x i in a solvable subgroup of G on which π γ is a representation. In this case the atomic decomposition becomes By choosing α = g and γ = 2g/p we recover Theorem 1 of Coifman and Rochberg [14] for Namely, there are functionals Here the functionals λ i differ from λ i by unimodular factors. For a parameter θ we can let γ = (2α + θg)/p. Then we get the expansion from Theorem 2 in [14] for general bounded domains. Namely, there are functionals Here the functionals λ i differ from h(w i ) − θg 2p λ i by unimodular factors.
The special case of the unit ball in C n was treated in [9] and this article follows the same strategy and we will usually use results from that article as far as the statements are stated in full generality or the generalization is trivial.

Background on Bergman spaces
For z ∈ D let x z ∈ G be such that x z · 0 = z. The element x z is not unique because x · z = (xk) · 0 for all k ∈ K. The following is well known but we give a short discussion of the proof as this will be used quite often in the following: Proof. a) We have J(e, z) = 1. Hence the cocylce relation gives The reproducing property of K(z, w) gives: But this can also be calculated as Replacing w by x · w and using that J( This follows easily from (b). Then Proof. We first note that h(u, 0) = h(0, v) = 1 for all u, v ∈ D. Hence by Lemma 3.1 part b: By part (c) in the above lemma we have Then take x = y to get This proves the statement.
Then the following holds: ( These estimates provide the following result about boundedness for an integral operator with kernel given by the modulus of the Bergman kernel. A more general result was proved by Bekolle and Temgoua Kagou (see Theorem II.7 in [4]).
Proof. This involves a standard trick based on Schur's lemma. Notice that for ǫ > 0 we have, by Hölder's inequality with 1/p + 1/q = 1, if qǫ > (r − 1)a/2. By Tonelli's theorem this equals which by Theorem 3.3 is less than . The restrictions for the parameters α and γ required by Theorem 3.3 can be rewritten to The proof of the wavelet characterization in Theorem 2.1 will be carried out using ideas from coorbit theory for projective representations as presented in [9]. For this purpose we will rephrase Theorem 3.4 in the form of boundedness of convolution operators on G.
For the remainder of this paper we normalize the Haar measure on G such that if f is For functions F, G on the group G we define convolution in the usual way: Theorem 3.5. The operator C γ : Proof. We already know that h(y −1 x) = h(x)h(y)/|h(x, y)| 2 , so C γ takes the form , so by Fubini's Theorem and the right-K-invariance of h we get In the second to last estimate we used the boundedness of the operator T . This shows that C γ is bounded on L P α−γp/2 (G) for the specified ranges of α,γ and p.

Proof of the first part of Theorem 2.1
We now turn to the proof of Theorem 2.1.
Proof. Lemma 2.1 in [20] states that Proof. First, if α > g −1, then by Lemma 4.1, there is a β > 2α/p + g −1 for which A p α ⊆ A 2 β . It is known that the monomials ψ m (z) = 1 z m A 2 β z m form an orthonormal basis for A 2 β , so Let P k denote the space of polynomials of homogeneous degree k, then f = k f k for f k ∈ P k , and f k is given by Taking the A 2 γ norm and using the triangle inequality, we get By Proposition 3.3 in [6] there are integers N 1 ,N 2 and a constant C, which only depend on β and γ, such that From this estimate we get The fact that there is a constant A and an integer N 3 for which finishes the proof that f k γ ≤ C(1 + k) N for some integer N and for some constant C which do not depend on k. Therefore, f is in S * γ . To prove the first part of Theorem 2.1 let ψ = 1 and consider the representation π γ . Then the wavelet coefficients with ψ = 1 and f ∈ S * γ become We see that and from this it follows that if W γ ψ (f ) is in L p α−γp/2 (G) then f ∈ A p α . If f ∈ A p α , then by Lemma 4.2 f is in S * γ . Then equation (4.2) tells us that W γ ψ (f ) is in L p α−γp/2 (G). This finishes the proof of the first part of Theorem 2.1.

Proof of the second part of Theorem 2.1
Before we can prove the second part of the theorem, we need to better understand the growth of wavelet coefficients for vectors in S γ . The following gives the estimates of wavelet coefficients which turn out to be a crucial part of our proof.
Using this result and Theorem 3.3 it is possible to show that |W u (v)| ≤ Ch γ/2−ǫ for any ǫ > (r − 1)a/2. We expect that this inequality can be verified for all ǫ > 0. This would enable us to answer some of the questions posed in Remark 2.2.
Proof. We first assume that u ∈ S γ and v is a polynomial. Let us calculate the wavelet coefficient for Since K γ is the reproducing kernel for A 2 γ and since ψ m = 1 This yields Since v is a polynomial and z → u(x −1 · z) is holomorphic, we get that the sum over m is finite, i.e.
where M is a finite index set (which can be chosen independently of x). By Lemma 1.3 and Proposition 3.3 in [6] we get that u is bounded. Therefore, there is a constant C m for which By the finiteness of M we can therefore derive that for x ∈ G we have Now assume that v ∈ S γ is not a polynomial. For x, y ∈ G equation (4.1) gives This function is well-defined for y in G, and So far we have not needed to introduce a projective representation of G along with its cocycle. We choose to introduce them now, since subsequent arguments are easily carried out based on established knowledge about square integrable projective representations and twisted convolution. In particular we will prove the second part of Theorem 2.1 by using twisted convolution to swap the vector ψ = 1 in W γ ψ (f ) by an arbitrary vector in S γ . Let ρ γ be a projective representation of G corresponding to the representation π γ of G, and let σ γ be the corresponding cocycle for ρ γ . We retain the notation W γ ψ : S * γ → B(G) for the wavelet transform W γ ψ (f )(x) = f, ρ γ (x)ψ , and note that it agrees with the previous notation up to a unimodular factor. Also define twisted convolution of f, g by when the integral makes sense.
For a function F on G and a vector X ∈ g define the derivative Lemma 5.5. There is a sequence of smooth compactly supported functions ψ n : G → [0, 1] such that ψ n+1 ≥ ψ n and for any finite collection X 1 , X 2 , . . . , X N ∈ g there is a constant C N such that for every n we have X 1 X 2 . . . X N ψ n ∞ ≤ C N .
Proof. We can take ψ ∨ n (x) = ψ n (x −1 ) to be partial sums of a partition of unity as contstructed on p. 329 in [23].
Proposition 5.6. Let ψ, φ and η be in S γ and assume that Proof. First, note that if W γ ψ (f ) is in L p α−γp/2 (G), then by Lemma 5.4 the twisted convolution is defined in terms of an integrable function. Therefore we can employ the Lebesgue Dominated Convergence Theorem to get that where ψ n ∈ C ∞ c (G) is a sequence of functions which are equal to one on growing compact sets whose union is G and which satisfy 0 ≤ ψ n (x) ≤ 1 for all x ∈ G. The equation (5.1) rewrites to Replace ρ γ (x)φ by a general smooth vector ξ and define the smooth compactly supported function Ψ n (y) = ψ n (y) ρ γ (y)η, ξ , then G ψ n (y) f, ρ γ (y)ψ ρ γ (y)η, ξ dy = G f, Ψ n (y)ρ γ (y)ψ dy.
Since Ψ n is smooth and compactly supported, by Theorem 3.27 in [21] the vector ρ γ (Ψ n )ψ := G Ψ n (y)ρ γ (y)ψ dy is a smooth vector, and In order to finish the proof it suffices to show that the latter expresson converges to c f, ξ where c = d −1 γ ψ, η . We will do so by demonstrating that the vectors ρ γ (Ψ n )ψ converge to cξ in S γ .
Remark 5.7. If one chooses to define twisted convolution of W ψ (f ) and W φ (η) using the expression (5.1), then the proof in fact shows that it equals d −1 γ η, ψ A 2 γ W γ φ (f ) regardless of the integrability condition. This could be a key observation for applying coorbit theory in this setting for dealing with the extended range of α.
We have gathered all the results needed to finish the proof of Theorem 2.1. First, we know that f ∈ A p α if and only if f ∈ S * γ and W γ 1 (f ) ∈ L p α−γp/2 (G). Employ Proposition 5.6 to show that for an arbitrary vector ψ ∈ S γ we have W γ ψ (f ) = CW 1 (f )#W ψ (1) for non-zero constant C. By Lemma 5.4 we get that . If, on the other hand, W γ ψ (f ) is in L p α−γp/2 (G), then Lemma 5.4 and Proposition 5.6 tell us that , where D is a non-zero constant. By the first part of Theorem 2.1 we get that f is in A p α . These calculations also show that the norms are equivalent, and the proof is done.

Proof of Theorem 2.3
This result relies on the fact that the space S γ is in fact the space of smooth vectors for the representation π γ (see [6]). Therefore the space S γ is invariant under the differential operator where X is in the Lie algebra g of G. This means that the left and right derivatives of a wavelet coefficient W γ u (v), where u and v are in S γ will correspond to wavelet coefficients Moreover, we know that L p α−γp/2 * |W γ u ′ (v ′ )| ⊆ L p α−γp/2 for the specified range of parameters when u ′ and v ′ are in S γ . Therefore, the work for this has already been done by establishing Theorem 3.5 and Corollary 5.3 while noticing that |W γ 1 (1)| = h γ/2 . Combining this with Theorem 8 from [9] we get the desired result for the specified range of α, γ and p.

The unbounded case
In this section we explain how the atomic decompositions for the unbounded realization of the domain can be obtained. The idea is simply to use the Caley transform to transform information from the bounded domain to the unbounded realization. In particular, the representation theory makes the transition from the bounded to the unbounded case clear. Let c : U → D be the Cayley transform from the irreducible unbounded symmetric domain U to the irreducible bounded symmetric domain D. Note that there exists an element c ∈ G C , where G C is the simply connected complex Lie group with Lie algebra g C such that c(z) = cz. The Bergman space A 2 (U) is defined as the holomorphic functions for which Here dz denotes the Euclidean measure on C n which gives the unit cube volume 1. The Bergman space is a reproducing kernel Hilbert space with reproducing kernel denoted K U satisfying Let J(c, w) denote the complex Jacobian of the mapping c : U → D, then C : A 2 (D) → A 2 (U) defined by Cf (z) = J(c, z)f (cz) is an isometry. Also, remembering that the Bergman kernel on the bounded realization is denoted K, we have J(c, z)J(c, w)K(cz, cw) = K U (z, w).
(2) One could also use the isometry C α,p : A p α (D) → A p α (U) to transfer the atomic decomposition from the bounded domain to the unbounded realization. In particular, if F ∈ A p α (U) and if ψ is in S γ , then This sum converges absolutely in A p α (D), and since π γ (x i )ψ ∈ S γ ⊆ A p α (D) it follows that Finally note that this equals Due to the factor J(c, z) 2α pg − γ g the atoms in this decomposition are not translates under τ γ of a single function. This makes this decomposition obtained in Theorem 7.2 more attractive.