On the theory of Bergman spaces on homogeneous Siegel domains

We consider mixed-norm Bergman spaces on homogeneous Siegel domains. In the literature, two different approaches have been considered and several results seem difficult to be compared. In this paper, we compare the results available in the literature and complete the existing ones in one of the two settings. The results we present are as follows: natural inclusions, density, completeness, reproducing properties, sampling, atomic decomposition, duality, continuity of Bergman projectors, boundary values, and transference.


Introduction
This paper deals with some spaces of holomorphic functions on a homogeneous Siegel domain.In order to illustrate the kind of spaces and problems we are going to consider, we begin with the simplest case.
Let C + := { z ∈ C : Im z > 0 } be the upper half-plane.We can think of C + as R + i(0, ∞), where (0, ∞) is the unique open (convex) cone in R. In several variables, the upper half-plane can be generalized to tube domains over convex cones.Let Ω be a convex open cone in R m .Then, the domain D = R m + iΩ in C m is called the tube domain over the cone Ω.If the group of linear transfomations of R m that preserve Ω acts transitively on Ω itself, then Ω is a homogeneous cone and the domain becomes itself homogeneous, that is, the group of biholomorphic self-maps of D (the automorphisms of D) acts transitively on D.
Another classical domain in several variables that extends the definition and some of the main features of C + is the so-called Siegel upper half-space.Consider again the cone (0, ∞) ⊆ R and the hermitian quadratic map on Then, the Siegel upper half-space is the domain U in C n × C U := (z, ζ) : Im z − |ζ| 2 ∈ (0, ∞) .The homogeneous Siegel domains are then introduced as follows -we refer to Section 2 for complete definitions.Let a homogeneous cone Ω ⊂ R m and a suitable hermitian quadratic map Φ : C n → C m be given.Then, the homogeneous Siegel domain (again, cf.Section 2 for definitions).Notice that if n = 0, then D is the tube domain over the given cone Ω.
On a homogeneous Siegel domain D as above, various (mixed norm) weighted Bergman spaces have been considered in the literature.On the one hand, in [30] (for the upper half-plane) and [15] (for the general case), the following mixed norm weighted Bergman spaces are considered: A p,q s := f ∈ Hol(D) : h → ∆ s Ω (h) f h L p (N ) L q (νΩ ) < ∞ , where ∆ s Ω are 'generalized power functions' on Ω (s ∈ R r ), ν Ω is 'the' invariant measure on Ω, N = C n ×R m and f h : (ζ, x) → f (ζ, x + iΦ(ζ) + ih).On the other hand, e.g. in [1,11,3,2,21,26,6,4,7], the following mixed norm weighted Bergman spaces are considered: where b is a suitable element of R r and L p,q (R m , Two parallel theories then arise, and different conventions have been adopted.For example, the definition of the spaces A p,q s suggests a natural comparison between the spaces A p,q s for a fixed s, which in turn highlights the role played by 'the' Bergman projector P s , namely the Bergman projector of the corresponding space A 2,2  s .On the other hand, the comparison of the spaces A p,q s for fixed s appears to be less natural, so that more general Bergman projectors are naturally investigated.Besides that, in the study of various properties of the spaces A p,q s (such as, for instance, the continuity of P s on the space L p,q s , which is defined as A p,q homogeneous if and only if there is a triangular 1 subgroup T + of GL(F ) which acts simply transitively on Ω, and for every t ∈ T + there is g ∈ GL(E) such that t • Φ = Φ • (g × g).In this case, any other triangular subgroup of GL(F ) with the same properties is conjugated to T + by an element of GL(F ) which preserves Ω.In addition, T + acts simply transitively on the right on Ω ′ , by transposition (cf.[35,Theorem 1]).We shall denote this latter action by λ • t, for λ ∈ Ω ′ and t ∈ T + ; we shall consequently write t • h instead of th for t ∈ T + and h ∈ Ω.We shall still denote by t • and • t the actions of t on F C and F ′ C , respectively, for every t ∈ T + .
2.1.Analysis on Ω.It is possible to describe the structure of T + and of its action on Ω using the theory of T -algebras, cf.[35], or the theory of (normal) j-algebras, cf.[29,32].In order to keep the exposition as simple as possible, we shall avoid a thorough description of the structure of T + and proceed axiomatically.We refer the reader to [15] for a more detailed treatment of the following considerations.We first observe that there are r ∈ N (called the rank of Ω) and a surjective homomorphism of Lie groups ∆ : T + → (R * + ) r , with kernel [T + , T + ], such that, if we fix base-points e Ω ∈ Ω and e Ω ′ ∈ Ω ′ and define (1) for every s ∈ C r and for every t ∈ T + , then ∆ s Ω (and ∆ s Ω ′ ) is bounded on bounded subsets if and only if Re s ∈ R r + (cf.[15,Lemma 2.34]).We shall further require that ∆(a • ) = (a, . . ., a) for every a > 0, where a • denotes the homothety of ratio a (which necessarily belongs to T + ).We remark explicitly that these conditions determine ∆ up to a permutation of the coordinates (in (R * + ) r ). 2 Consequently, we may apply the results of [15,Chapter 2] without (essential) changes, even if a different choice of T + and ∆ is made.Notice that ∆ s Ω and ∆ s Ω ′ extend to holomorphic functions on Ω + iF and Ω ′ + iF ′ , respectively, for every s ∈ C r (cf.[15,Corollary 2.25]).
When Ω is symmetric, that is, self-dual with respect to the scalar product of F , then the functions ∆ s considered in [22] coincide with the functions ∆ s Ω defined in (1) for an appropriate choice of ∆ (cf.[22, Chapter VI, § 3]); in particular, the 'determinant' polynomial concides with ∆ 1r Ω .Generally speaking, the works which deal with the case in which Ω is symmetric generally adhere to the conventions of [22], possibly with slightly different notation, whereas the works which deal with general homogeneous cones generally adhere to the conventions described above, possibly with different notation (for example, ∆ s Ω = Q s and ∆ s Ω ′ = (Q * ) s in the notation of [26,27,8,6]) To simplify the notation, we state the following definition.
Definition 2.1.We define two order relations on R r .On the one hand, we write s s ′ to mean s j s ′ j for every j = 1, . . ., r (equivalently, s ′ − s ∈ R r + ).On the other hand, we write s s ′ to mean s = s ′ or s j < s ′ j for every j = 1, . . ., r.
Thus, s ≺ s ′ (that is, s s ′ and s = s ′ ) if and only if s ′ − s ∈ (R * + ) r , that is, s j < s ′ j for every j = 1, . . ., r. Definition 2.2.We denote by H k the k-dimensional Hausdorff measure.There are d ≺ 0 and b 0 such that (2) ν This means that all the eigenvalues of every element of T + are real.Equivalently, there is a basis of F with respect to which every element of T + is represented by an upper triangular matrix, cf.[34]. 2 To see this fact, observe that, if ∆ ′ : T + → (R * + ) r is another homomorphism with the same properties, then there is A ∈ GL(R n ) such that log ∆ ′ = A log ∆.In addition, given s ∈ R r , both j s j log ∆ ′ j = j ( t As) j log ∆ j and j s j log ∆ j induce functions which are bounded above on the bounded subsets of Ω if and only if s ∈ R * + , so that t AR * + = R * + and therefore A must be the composition of a permutation of the coordinates and a diagonal dilation (x 1 , . . ., xr) → (λ 1 x 1 , . . ., λrxr), λ 1 , . . ., λr > 0. Since ∆(a • ) = ∆ ′ (a • ) = (a, . . ., a) for every a > 0, we then see that A must induce the identity on the line R1r, so that it must be a permutation of the coordinates.
are the unique measures on Ω, Ω ′ , and D (up to a multiplicative constant) which are invariant under all linear automorphisms of Ω and Ω ′ , and all biholomorphisms of D, respectively (cf.[ We observe explicitly that d = d and b = −q in the notation of [24,1,9], whereas d = −τ and b = −b in the notation of [27,26,8,6].In particular, there is no general agreement on the sign of d. [15, Definition 2.8] and the preceding remarks).We observe explicitly that m = (m 1 , . . ., m r ) and m ′ = (n 1 , . . ., n r ) in the notation of [9,27,26,8,6,7].Definition 2.6.For every s, s ′ ∈ C r such that Re s ≻ 1 2 m and Re s ′ ≻ 1 2 m ′ , we define Γ Ω (s) and for some constants c, c ′ > 0 which depend on the choice of e Ω and e Ω ′ .Definition 2.8.There are two uniquely determined holomorphic families (I s Ω ) s∈C r and (I s Ω ′ ) s∈C r of tempered distributions on F and F ′ , respectively, such that LI s

and Proposition 2.28]).
Remark 2.9.Notice that for a suitable constant c > 0 which depends on the choice of e Ω ′ (cf.[15,Proposition 2.30]).We observe explicitly that, when Ω is symmetric, then 1 r ∈ N Ω and the differential operator f → f * I −1r is simply differential operator associated with the determinant polynomial ∆ 1r Ω by means of the scalar product.This latter operator is often denoted by .In addition, if Ω is symmetric and irreducible, then for an appropriate choice of ∆.This latter condition completely determines ∆ in this case.

2.2.
Fourier Analysis on the Šilov Boundary.We now pass to the analysis of the Šilov boundary of D (cf.[28] for a more general treatment of this topic).We endow E × F C with the 2-step nilpotent Lie group structure whose product is given by so that W is a proper algebraic variety in F ′ since Φ is non-degenerate and Ω-positive.Then, for every λ ∈ F ′ \ W , the quotient of N modulo the central subgroup ker λ is isomorphic to a Heisenberg group (to R, if E = { 0 }), so that the Stone-Von Neuman theorem (cf., e.g., [23,Theorem 1.50]) ensures the existence of a unique (up to unitary equivalence) irreducible continuous unitary representation π λ of N in some Hilbert space H λ such that π λ (0, x) = e −i λ,x for every x ∈ F .One then has the Plancherel identity (cf.[15, Corollary 1.17 and Proposition 2.30]): , where c > 0 is a suitable constant (which depends on the choice of e Ω ′ ) and L 2 (H λ ) denotes the space of Hilbert-Schmidt endomorphisms of H λ .Note that ∆ −b Ω ′ is positive on Ω ′ and extends to a polynomial on F ′ , so that the above formula is meaningful (cf.[15, Proposition 2.30]).

2.3.
The CR Structure of N .For every v ∈ E, denote by Z v the left-invariant vector field on N which induces the Wirtinger derivative 1  2 (∂ v − i∂ iv ) at (0, 0).Then, the Z v , for v ∈ E, induce a subbundle of the complexified tangent bundle of N which endows N with the structure of a CR manifold (cf.[10,Section 7.4]).In particular, a distribution u on N is said to be CR if Z v u = 0 for every v ∈ E (cf.[10, Sections 9.1 and 17.2]).Note that an element f of L 2 (N ) is CR is and only if for almost every λ ∈ F ′ \ W , where Λ + is the interior of the polar of Φ(E) (that is, and P λ,0 is an orthoprojector of rank one in H λ , for every λ ∈ F ′ \ W (cf., e.g., [28] or [15,Proposition 1.19] and [14, Proposition 2.6]).
2.4.Metrics.We endow D with a complete Riemannian metric which is invariant under the action of affine biholomorphisms (for example, the Bergman metric is complete and invariant under all biholomorphisms of D, cf.[15, Proposition 2.44]), and the associated distance d.Since the balls with respect to d will only be used for bounded radii, it will not matter which distance is chosen, as long as it satisfies the preceding conditions.
We endow Ω with the Riemannian metric induced by that on D by means of the submersion ρ (interpreted as the projection of D onto its quotient modulo the action of N ), and Ω ′ with the metric induced by the diffeomorphism Ω ∋ t • e Ω → e Ω ′ • t −1 ∈ Ω ′ .We denote by d Ω and d Ω ′ the corresponding distances, and by B Ω (h, R) and B Ω ′ (λ, R) the correspondings balls of centre h ∈ Ω and λ ∈ Ω ′ , respectively, and radius R > 0. Notice that also in this case one may choose general complete T + -invariant Riemannian distances without (essentially) compromising the results which follow.Nonetheless, the relationships between d and d Ω will be useful in some places (such as in the definition of lattices given below).
Analogously, we endow E × Ω with the the Riemannian metric induced by the one on D by means of the submersion interpreted as the projection of D onto its quotient modulo the action of he centre F of N .We denote by d E×Ω the corresponding distance, and by B E×Ω ((ζ, h), R) the corresponding ball of centre (ζ, h) ∈ E × Ω and radius R > 0.
We observe explicitly that both N and its centre F are normal subgroups of the group G Aff of affine automorphisms of D (cf.[25,Proposition 2.1]).Hence, d Ω and d E×Ω are (G Aff /N )-and (G Aff /F )-invariant, respectively.In particular, d Ω and d Ω ′ are T + -invariant, while d E×Ω is invariant under the affine automorphisms of the form 2.5.Lattices.By a (δ, R)-lattice on Ω, with δ > 0 and R > 1, we mean a family (h k ) k∈K of elements of Ω such that the balls B Ω (h k , δ) are pairwise disjoint while the balls B Ω (h k , Rδ) cover Ω.We define lattices on Ω ′ and E × Ω analogously.Notice that every maximal family of elements of Ω whose mutual distances are 2δ is necessarily a (δ, 2)-lattice (and conversely), so that (δ, 2)-lattices on Ω, Ω ′ , and E × Ω always exist.By an N -(δ, R)-lattice on D, with δ > 0 and R > 1, we mean a family (ζ j,k , z j,k ) j∈J,k∈K of elements of D such that the balls B((ζ j,k , z j,k ), δ) are pairwise disjoint, the balls B((ζ j,k , z j,k ), Rδ) cover D, and there is a (δ, R)-lattice (h k ) k∈K on Ω such that ρ(ζ j,k , z j,k ) = h k for every j ∈ J and for every k ∈ K.
By an F -(δ, R)-lattice on D, with δ > 0 and R > 1, we mean a family (ζ k , z j,k ) j∈J,k∈K of elements of D such that the balls B((ζ k , z j,k ), δ) are pairwise disjoint, the balls B((ζ k , z j,k ), Rδ) cover D, and there is a for every j ∈ J and for every k ∈ K.
By a modification of the previous argument, one may show that N -and F -(δ, 4)-lattices always exist on D (cf.[15, Lemma 2.55]).
2.6.The Associated Tube Domain.We denote by the tube domain associated with Ω.Given a function f on D, we define for every h ∈ Ω, and for every ζ ∈ E and for every h ∈ Ω.

Statement of the Main Comparison Results
We now introduce the different definitions of mixed-norm Bergman spaces.In [30,15], mixed-norm weighted Bergman spaces are defined as On the one hand, this definition highlights the role played by the Šilov boundary of D and gives rise to the usual Hardy spaces when q = ∞ and s = 0 (that is, ∆ s Ω = 1).In particular, the non-commutative Fourier analysis on N comes into play.On the other hand, the weight ∆ s Ω • ρ is considered as a multiplier of the function, and not of the measure, and the 'base measure' is chosen in such a way that it induces the invariant measures on N and Ω.When q = ∞, this allows to treat a whole class of spaces which would not appear otherwise, and which play a relevant role in the duality theory of the spaces A p,q s (D) when q 1.In [26,6] (to cite only a few), mixed norm weighted Bergman spaces are defined as (s+b/2)/q (TΩ ) L q (E) < ∞ .On the one hand, this definition highlights the role played by the contre F of the Šilov boundary of D, so that the usual (commutative) Fourier analysis on F comes into play.In addition, this definition also allows to think D as the union of the translates (ζ, iΦ(ζ))+ T Ω of the tube domain T Ω (identified with { 0 }× T Ω ⊆ D), so that some of the analysis on A p,q s (D) may be reduced to a simpler analysis on A p,q s (T Ω ).On the other hand, the weight ∆ s Ω •ρ is considered as a multiplier of the 'base measure' •ρ)•ν D , and not of the function.In this way the self-adjoint projector of L 2,2 s (D) (defined as A 2,2 s (D), but allowing f to be a meausurable function modulo negligible functions) onto A 2,2 s (D) is highlighted as the 'canonical choice' when looking for a projector of L p,q s (D) onto A p,q s (D) We mention that A p,∞ s (D) = A p,∞ 0 (D) for every s ∈ R r .Because of this fact, the case q = ∞ is somewhat pathological and seldom considered.For similar reasons, the duality theory for the space A p,q s (D), when q 1, is treated separately (cf., e.g., [1]).
We also observe that for every measurable function g : N → C.
In [15], for notational convenience the corresponding integral operators are based on B s rather than K s , so that the operators are considered.Then, The following result is a consequence of [15, Proposition 3.13] and Proposition 4.7.The first assertion is contained in [30,Theorem 3.1] when D = C + .Proposition 3.4.Take p, q ∈ (0, ∞] and s, s ′ ∈ R r .If: then P s ′ f = f for every f ∈ A p,q s (D).3.3.Sampling.The following sampling theorems are consequences of [15,Theorem 3.22] and Theorem 4.9, where more precise versions of these results are proved.The second assertion is contained in [3,Theorem 5.6] when n = 0, p = q, and Ω is symmetric, in [8, Theorem 5.2] when p = q, and in [7,Theorem 3.3] when n = 0 and Ω is symmetric.We denote by ℓ p,q (J, K) the space of λ ∈ C J×K such that λ j,k ℓ p (J) ℓ q (K) < ∞, with some abuse of notation.Theorem 3.5.Take p, q ∈ (0, ∞], s ∈ R r and R 0 > 1.Then, there is δ 0 > 0 such that, for every induces an isomorphism of A p,q s (D) onto a closed subspace of ℓ p,q (J, K), and such that, for every induces an isomorphism of A p,q s (D) onto a closed subspace of ℓ p,q (J, K).Here we mention that the transpose of the sampling maps defined above is often considered an atomic decomposition map, especially when the duals of A p,q s (D) and A p,q s (D) may be identified with , respectively, for some s ′ .See [15,18] and Subsection 4.3 and Theorem 4.28 for more information.Since, in particular, the problem of determining the validity of the aforementioned atomic decomposition and duality is equivalent (when p, q ∈ [1, ∞]) to the problem of determining the range of boundedness of the Bergman projectors, we shall only discuss the latter problem in this section.

In addition, S (ζ,z) ∈ B
A similar result holds for the spaces B s p,q (N , Ω) as well (cf.Lemma 4.22 and the remarks following the statement of Proposition 4.25).Nonetheless, because of the pathological behaviour of the spaces B s p,q (N , Ω) for q = ∞, it is not possible to state the analogous result here without defining additional spaces.We therefore refer the reader to the aforementioned results for more precise information.
s−(b+d)/p (D), and denote by A p,q s (D) its image, endowed with the corresponding topology.If q < ∞, given s ≻ 1 2 b + q p d + q 2q ′ m ′ , we define a continuous linear mapping . We denote by A p,q s (D) the image of B −s p,q (N , Ω) under E, endowed with the corresponding topology.Notice that (Eu) h → u in S ′ Ω ′ (N ) for h → 0, h ∈ Ω ′ , for every u ∈ B −s p,q (N , Ω) and for every u ∈ B −s p,q (N , Ω) (cf.[15, Theorem 5.2], [13, Proposition 7.12], and Proposition 4.23), so that E is one-to-one and B −s p,q (N , Ω) and B −s p,q (N , Ω) are the spaces of boundary values of A p,q s (D) and A p,q s (D), respectively (when defined).

with equality in the second inclusion if
We also have transference results (cf.[18,Theorem 6.3] and Proposition 4.24).Proposition 3.12.Take p, q ∈ (0, ∞] and s ∈ R r .Then the following hold: 3.5.Bergman Projectors.Concerning the boundedness of Bergman projectors, we have the following results.
Proposition 3.13.Take p, q ∈ [1, ∞] and s, s ′ ∈ R r .Then, the following hold: (1) if s ′ ≺ b + d − 1 2 m and P s ′ induces a continuous linear projector of L p,q s (D) onto A p,q s (D), then: and s ≻ 1 2q m or q = ∞ and s 0; 2 (m − b), and P s ′ induces a continuous linear projector of L p,q s (D) onto A p,q s (D), then: ), and P s induces a continuous linear projector of L p,q s (D) onto A p,q s (D), then: There are also transference results (cf.[18,6] and Corollary 4.29).
Proposition 3.14.Take p, q ∈ [1, ∞] and s, s ′ ∈ R r .Then, the following hold: 2 m and P s ′ induces a continuous linear projector of L p,q s (D) onto A p,q s (D), then P s ′ induces a continuous linear projector of L p,q s−b/p (T Ω ) onto A p,q s−b/p (T Ω ); 2 (m − b) and P s ′ +b/2 induces a continuous linear projector of L p,q s+b/2 (T Ω ) onto A p,q s+b/2 (T Ω ), then P s ′ induces a continuous linear projector of L p,q s (D) onto A p,q s (D).The following result is a consequence of [15, Corollary 5.27] (cf.also [18,Corollary 4.7]), Proposition 4.24, and Theorem 4.28.The second assertion is contained in [2, Theorem 1.9] when p, q 1, s = s ′ , Ω is symmetric, and s ∈ Rd, and in [21, Corollary 1.4] when p, q 1, s = s ′ , and Ω is symmetric.Theorem 3.15.Take p, q ∈ [1, ∞] and s, s ′ ∈ R r .Then, the following hold: (1) if: then P s ′ induces a continuous linear projector of L p,q s (D) onto A p,q s (D); (2) if q < ∞ and: then P s ′ induces a continuous linear projector of L p,q s (D) onto A p,q s (D).In particular, if s = s ′ ≻ 1 2 (m − b) and q ′ s (p) < q < q s (p), then P s ′ induces a continuous linear projector of L p,q s (D) onto A p,q s (D) (cf. the remarks after the statement of Theorem 3.10).

An Auxiliary Space
For technical reasons, we shall consider the spaces A p,q s (TΩ ) L q (E) < ∞ and A p,q s,0 (D) := Hol(D) ∩ L p,q s,0 (D), where L p,q s,0 (D) denotes the closure of C c (D) in L p,q s (D) (defined as A p,q s (D) replacing Hol(D) with the space of measurable functions modulo negligible functions).
Note that, as the proof shows, if p = q and E = { 0 }, then A p,q s (D) ⊆ A p,q s (D) and A p,q s (D) ⊇ A p,q s (D).Proof.The first assertion is clear.As for what concerns the second assertion, it is clear that A p,q s (D) = A p,q s (D) if p = q (by Fubini's theorem) and if E = { 0 }.Conversely, assume that p = q and that E = { 0 }, so that b = 0. Observe that, given t ∈ T + and g ∈ GL(E) such that t so that, letting t → ∞, we see that the norms on the quasi-Banach spaces A p,q s (D) and A p,q s (D) cannot be comparable.The assertion follows from the open mapping and the closed graph theorems.Proposition 4.3.Take p 1 , p 2 , q 1 , q 2 ∈ (0, ∞] and s 1 , s 2 ∈ R r .If (D) and A p1,q1 s1,0 (D) ⊆ A p2,q2 s2,0 (D).Proposition 4.5.Take p 1 , p 2 , q 1 , q 2 ∈ (0, ∞], and s 1 ,
By [15, Corollary 2.49], there is a constant C > 0 such that, for every f ∈ Hol(D) and for every h ∈ Ω, Then, Jensen's inequality (with exponent q ℓ ) leads to the conclusion.Proof of Theorem 4.9.For the sake of simplicity, we shall generally present the computations as if p, q < ∞.We leave to the reader the (purely formal) modifications which are necessary when max(p, q) = ∞.Throughout the proof, for every t ∈ T + we shall denote by g t an element of GL(E) such that t•Φ = Φ•(g t ×g t ).
Step III.Now, take f ∈ Hol(D) and assume that S + f ∈ ℓ p,q 0 (J, K) (resp.S + f ∈ ℓ p,q (J, K)), and let us prove that f ∈ A p,q s,0 (D) (resp.f ∈ A p,q s (D)).Observe first that and there is a constant C ′′ 2 > 0 (cf.[15,Corollary 2.49]) such that )-lattice on D, then there are two mappings ι 1 : K ′ → K and ι 2 : and observe that, by [15,Corollary 2.49] and the preceding remarks, there is a constant C ′ > 0 such that for every f ∈ Hol(D), for every j ′ ∈ J ′ and for every k ′ ∈ K ′ .In addition, there is N ′ ∈ N such that the fibres of ι 1 and 2 is a (δ, R ′ )-lattice on Ω for some/every k ′ 1 ∈ K ′ 1 (argue as in the proof of [15, Lemma 2.55]).
Step V. Take f ∈ M s ′ (D) such that S − f ∈ ℓ p,q (J, K) and let us prove that Cδ m/p+(2n+m)/q S − f ℓ p,q (J,K) for a suitable constant C > 0 (depending only on δ − and R 0 ), provided that δ − is sufficiently small.Observe first that, by step IV, up to replacing R with R+ 8, we may assume that only depends on k ′′ (so that we also write h k ′′ instead of h (k ′ ,k ′′ ) ), and that (h k ′′ ) is a (δ, R)-lattice in Ω. Observe that, for every (j, k) ∈ J × K, we may find Now, [15,Lemmas 3.24 and 3.25] imply that there are R ′ 1 ∈ (0, R ′ 0 ] and C 3 > 0 such that, for every j ∈ J, for every k ∈ K, for every (ζ, x) ∈ N , and for every h where
and, for every λ ∈ C (J×K) , We shall first prove that λ → Ψ + (λ) L p,q s (D) is a continuous quasi-norm on ℓ p,q (J, K).Arguing as in [15, Proposition 3.32] and using Proposition 4.6, this will prove that Ψ induces continuous linear mappings ℓ p,q 0 (J, K) → A p,q s,0 (D) and ℓ p,q (J, K) → A p,q s (D).We shall then prove that these mappings are onto and have continuous linear sections.
Step I. Assume first that q p 1.Then, for every ζ ∈ E and for every h ∈ Ω, In addition, [15,Lemma 2.39] shows that there is a constant C 1 > 0 such that for every (j, k) ∈ J × K, for every ζ ∈ E, and for every h ∈ Ω.Therefore, using the subadditivity of the mapping x → x q/p on R + , Now, [15, Corollary 2.22 and Lemma 2.32] imply that there is a constant for every k ∈ K. Hence, 1/q 2 λ ℓ p,q (J,K) , whence our claim in this case.
Step II.Assume, now, that q p 1.For every Observe that by the computations of step I and [15, Corollary 2.49], there is a constant C ′ 1 > 0 such that where λ ′ = j |λ j,k | p k .Hence, by Minkowski's integral inequality and Young's inequality, and by [15,Lemma 2.32], , then it is clear that f L q/p (νΩ ) ν Ω (B Ω (e Ω , δ)) p/q λ ′ L q/p (K) = ν Ω (B Ω (e Ω , δ)) p/q λ p L p,q (J,K) .Then, [15,Lemma 3.35] implies that there is a constant C 3 > 0 such that which completes the proof of our claim in this case.
Step IV.Finally, assume that p 1 q.For every (j, k) h k , and define As in step III, one may show that Ψ ′′ induces a continuous linear mapping of ℓ p,q (J, K) into ℓ q (K; L p (F )).
In addition, by means of [15,Theorem 2.47] we see that there is a constant C 6 > 0 such that for every λ ∈ ℓ p,q (J, K), for every (ζ, h) ∈ E × Ω, and for every x ∈ F .Then, by Minkowski's inequality, Young's inequality, and [15, Lemma 2.39], there is a constant C ′ 6 > 0 such that for every λ ∈ ℓ p,q (J, K) and for every (ζ, h) ∈ E × Ω.Then, by the subadditivity of the mapping x → x q on R + , Now, by homogeneity, for every k ∈ K, and the last integral is finite by [15,Corollary 2.22 and Lemma 2.32].Therefore, there is a constant C 7 > 0 such that C 7 Ψ ′′ (|λ|) ℓ q (K;L p (F )) , whence our claim also in this case.
Step V. Put a well-ordering on J × K, and define for every (j, k) ∈ J × K, so that (U j,k ) (j,k)∈J×K is a Borel measurable partition of D (since J and K are countable).In addition, define c j,k := cν D (U j,k ) for every (j, k) ∈ J × K, where c > 0 is defined so that for every (j, k) ∈ J × K.Then, define so that Theorem 4.9 shows that S is well defined and continuous, and maps A p,q s,0 (D) into ℓ p,q 0 (J, K).Define S ′ := Ψ S.Then, Proposition 4.7 implies that, for every f ∈ A p,q s (D), Hence, [15, Theorem 2.47, Corollary 2.49, and Lemma 3.25] imply that there are R ′ 0 > 0 and C 8 > 0 such that for every j ′ ∈ J ′ and for every k ′ ∈ K ′ .Hence, Theorem 4.9 and the preceding steps show that there is a constant s (D) .Take δ 0 > 0 so that C 10 δ 0 1 2 , and assume that δ δ 0 .Then, for every k ∈ N, so that j∈N (I − S ′ ) j induces well defined endomorphisms of A p,q s,0 (D) and A p,q s (D), which are inverses of S ′ .Hence, induces well defined and continuous linear mappings from A p,q s,0 (D) into ℓ p,q 0 (J, K) and from A p,q s (D) into ℓ p,q (J, K), and Ψ Ψ ′′′ = S ′ j∈N (I − S ′ ) j = I.The proof is complete.Arguing as in [15,Proposition 3.37], one may prove the following result.Proposition 4.14.Take p, q ∈ (0, ∞] and s, s ′ ∈ R r such that property (L ′ ) p,q s,s ′ ,0 holds.Then, the sesquilinear mapping F ) such that the following hold: (1) for every u ∈ S ′ Ω ′ (N ), for every ζ ∈ E, and for every ϕ ∈ S(F Proof.Fix ϕ ∈ S(F ′ ) supported in Ω ′ , and define ψ := F −1 N (ϕ) and ψ ′ := F −1 F (ϕ).Then, π λ (ψ) = ϕ(λ)P λ,0 and π λ (δ 0 ⊗ ψ ′ ) = ϕ(λ)I for every λ ∈ Λ + , so that so that, by the arbitrariness of ϕ, the mapping is complete, the first assertion follows, as well as (1).Assertion ( 2) is a consequence of the fact that the mapping Lemma 4.16.Take p, q, p 2 , q 2 ∈ (0, ∞] with p p 2 and q q 2 , and a bounded subset For every ψ ∈ B and for every t ∈ T + , define ψ for every u ∈ S ′ (N ), for every ϕ, ϕ ′ ∈ B, and for every t, t ′ ∈ T + . Proof.
Step I.It will suffice to prove the first assertion when p 2 = q 2 = ∞ and t is the identity of T + , by Hölder's interpolation and homogeneity.Arguing by approximation as in the proof of [15,Corollary 4.7], we may further assume that u ∈ S(N ).Then, set ℓ := min(1, p, q) and take τ ∈ S Ω ′ (N ) so that ψ * τ = ψ for every ψ ∈ B, and observe that F,E) τ L (p/ℓ) ′ ,(q/ℓ) ′ (F,E) for every (ζ, x) ∈ N , whence the first assertion.
Definition 4.17.Take p, q ∈ (0, ∞] and s ∈ R r .Take a (δ, R)-lattice (λ k ) k∈K in Ω ′ for some δ > 0 and some R > 1, and fix a bounded family . Then, we define Bs p,q (N , Ω) (resp.B s p,q (N , Ω)) as the space of the u ∈ S ), endowed with the corresponding topology. 5n particular, B s p,p (N , Ω) = B s p,p (N , Ω) and Bs p,p (N , Ω) = Bs p,p (N , Ω) for every p ∈ (0, ∞] and for every s ∈ R. We now propose a different interpretation of B s p,q (N , Ω) which is particularly useful in certain situations.
We observe explicitly that in [15,Corollary 5.11] the assumption s ≻ 1 p (b + d) + 1 2q ′ m ′ is redundant (as the assumption s ≻ 1 q b + 1 p d + 1 2q ′ m ′ would be redundant in (6) above), as it is implied by the condition s ≻ We also mention that, if r = 2 (so that Ω is isomorphic to either a quadrant or a Lorentz cone), then combining [6, Theorems 6.6 and 6.8] (the latter being a consequence of [12,Theorem 1.2]) with [15, Theorem 5.10] (cf.also [18, remarks following Remark 2.6]), we see that A p,q s (T Ω ) = A p,q s (T Ω ) and A p,q s,0 (T Ω ) = A p,q s,0 (T Ω ) if and only if By transference, under the same condition we also have A p,q s (D) = A p,q s (D) and A p,q s,0 (D) = A p,q s,0 (D).
The proof is analogous to that of [18,Corollary 4.7].
In particular, we have the following transference result.
Corollary 4.29.Take p, q ∈ [1, ∞] and s, s ′ ∈ R r .Then, the following hold: • if s ′ ≺ d − 1 2 m and P s ′ induces a continuous linear projector of L p,q s,0 (T Ω ) onto A p,q s,0 (T Ω ) (resp. of L p,q s (T Ω ) onto A p,q s (T Ω )), then P s ′ +b induces a continuous linear projector of L p,q s,0 (D) onto A p,q s,0 (D) (resp. of L p,q s (D) onto A p,q s (D));

Concluding remarks
On the one hand, we have compared two parallel theories of mixed norm Bergman spaces on homogeneous Siegel domains.On the other hand, we have extended part of the theory for the spaces A p,q s to the spaces A p,q s .In doing this, we hope we have shed some light on the technically demanding subject of function theory on such domains.We believe that this is a lively area of research, that in recent times has drawn the interest of many scholars.We mention that the Šilov boundary of D naturally appear in the extension of the Paley-Wiener and Bernstein spaces of entire functions to higher complex dimensions, see in particular [14,19,20].

1 2 m
and P s ′ induces a continuous linear projector of L p,p s,0 (D) onto A p,p s,0 (D) (resp. of L p,p s (D) onto A p,p s (D)), then P s ′ induces a continuous linear projector of L p,p s−b/p,0 (T Ω ) onto A p,p s−b/p,0 (T Ω ) (resp. of L p,p s−b/p (T Ω ) onto A p,p s−b/p (T Ω )).