Poly-Bergman Type Spaces on the Siegel Domain: Quasi-parabolic Case

We introduce poly-Bergman type spaces on the Siegel domain Dn ⊂ Cn, and we prove that they are isomorphic to tensorial products of one-dimensional spaces generated by orthogonal polynomials of two kinds: Laguerre polynomials and Hermite type polynomials. The linear span of all poly-Bergman type spaces is dense in the Hilbert space L(Dn, dμλ), where dμλ = (Im zn − |z1|2 − · · · − |zn−1|2)λdx1dy1 · · · dxndyn, with λ > −1.


Introduction
In this paper we generalized the concept of polyanalytic function on the Siegel domain D n ⊂ C n , which is the unbounded realisation of the unit ball B n ⊂ C n .
For the Bergman space A 2 λ (D n ) of the Siegel domain D n , the analogues of the classical Bargmann transform and its inverse for five different types of commutative subgroups of biholomorphisms of D n were constructed in Quiroga- Barranco and Vasilevski (2007).In particular, for the parabolic case they found an isometric isomorphisms which is the Bargmann type transform, where In this work polyanalytic function spaces are defined via the complex structure of C n induced by the tangential Cauchy-Riemann equations given for the Heisenberg group Boggess (1991).Let L be (l 1 , . . ., l n ) ∈ N n .The poly-Bergman type space of D n , denote by A 2 λL (D n ) or simply by A 2 λL , is the subspace of L 2 (D n , dμ λ ) consisting of all L-analytic functions, i.e., functions that satisfy the equations where, as usual, Functions in A 2 λL will be also called polyanalytic functions.Anti-polyanalytic functions are just complex conjugation of polyanalytic functions, but they constitute a linearly independent space.For L = (l 1 , ..., l n ) ∈ N n , we define the anti-poly-Bergman type space Ã2 λL (D n ) (or simply Ã2 λL ) as the subspace of L 2 (D n , dμ λ ) consisting of all L-anti-analytic functions, i.e., functions satisfying the equations We define the spaces of true-L-analytic and true-L-anti-analytic functions as where A 2 λS = Ã2 λS = {0} if S N n , and {e k } n k=1 stand for the canonical basis of R n .The main results obtained in this work go as follows: 1) The space L 2 (D n , dμ λ ) admits the decomposition 2) There exists an unitary operator , where L l n −1 is the one-dimensional space generated by the Laguerre function of degree l n − 1 and order λ, and K ± (L) is the subspace of l 2 (Z n−1 ) ⊗ L 2 (R n−1 + , rdr) consisting of all sequences {c m (r)} Z n−1 such that c m belongs to a finite dimensional space generated by Hermite type functions.

CR Manifolds
For a smooth submanifold M of C n , recall that T p (M) is the real tangent space of M at the point p.In general, T p (M) is not invariant under the complex structure map J for T p (C n ).For a point p ∈ M, the complex tangent space of M at p is the vector space This space is sometimes called the holomorphic tangent space.Using the Euclidian inner product on T p (R 2n ), denote by X p (M) the totally real part of the tangent space of M which is the orthogonal complement of H p (M) in T p (M).We have that T p (M) = H p (M) ⊕ X p (M) and J(X p (M)) is trasversal to The complexifications of T p (M), H p (M) and X p (M) are denoted by T p (M) ⊗ C, H p (M) ⊗ C and X p (M) ⊗ C, respectively.The complex structure map J on T p (R 2n ) ⊗ C restrict to a complex structure map on is the direct sum of the +i and −i eigenspace of J which are denoted by H 1,0 p (M) and H 0,1 p (M), respectively.The following result establishes the form of the basis of H p (M).It also provides an expression for the generators of H p (M).We refer to Boggess (1991) for its proof.0) and Dh(0) = 0.A basis for H 1,0 p (M) near of the origin is given by If the graphing function h of M is independient of the variable x, then the local basis of H 1,0 p (M) has the following simple form We refer to Example 7.3-1 of Boggess (1991) for the details on the following construction of the Heisenberg group, which use the Equation (1).For the real hypersurface in C n defined by the generators for H 1,0 (M) are given by and the generators for H 0,1 (M) are given by (3)

Cauchy-Riemann Equations for the Siegel Domain
Let We often rewrite z as (z , z n ), where z = (z 1 , ..., z n−1 ).On the other hand, the usual norm in C n is denoted by | • |.In the Siegel domain we consider the weighted Lebesgue measure Recall now the well known weighted Bergman space A 2 λ (D n ), defined as the space of all holomorphic functions in and the unitary operator where dη λ (w) = v λ dμ(w).
Our aim is to introduce poly-Bergman type spaces in the Siegel domain, and then realize them in the space L 2 (D, dη λ ) in order to apply Fourier transform techniques for their study.We start with the image space A 0 (D) = U 0 (A 2 λ ), which consists of all functions ϕ(z , u, v) = (U 0 f )(w) satisfying the equations For functions satisfying this last equation, the first type equation in ( 4) can be rewritten as These kind of equations were used in Quiroga- Barranco and Vasilevski (2007), and without any restriction on ϕ, they proved to be more usefull than the first type of equations in (4), as explained right now.At first stage, our aim was to introduce poly-Bergman type spaces such that they densely fill the space L 2 (D n , dμ λ ), we additionaly required that such poly-Bergman type spaces be isomorphic to tensorial products of L 2 -spaces.Thus, following the techniques given in Quiroga- Barranco and Vasilevski (2007), equations ( 5) gave positive results for our porpuse.
In this way the differential operators given in (3) were found, and they certainly satisfy Obviously, a continuous function f is holomorphic in D n if and only if We will use the operators Λ k 's to define the first class of poly-Bergman type spaces, i.e., a certain class of polyanalytic function spaces.
On the other hand, the differential operators ∂/∂z k (k = 1, ..., n − 1) are used to define anti-analytic function spaces, but they can be replaced by the operators given in (2).By the way, In addition we must consider As expected, we use the operators Λ k 's to define anti-polyanalytic function spaces.

Orthogonal Polynomials Required
We will prove that poly-Bergman type spaces are isomorphic to tensorial products of one-dimensional spaces generated by orthogonal polynomials of two kinds.The first one is the set of Laguerre polynomials of order λ: L λ j (y) := e y y −λ j! d j dy j (e −y y j+λ ), j = 0, 1, 2, ...

Poly-Bergman Type Spaces
For L = (l 1 , ..., l n ) ∈ N n , we define the poly-Bergman type space A 2 λL as the subspace of L 2 (D n , dμ λ ) consisting of all functions f satisfying the equations Let {e j } n j=1 be the canonical basis of R n .We define the space of true-L-analytic functions as λL ) ⊂ L 2 (D, dη λ ) in order to apply Fourier techniques in the study of the poly-Bergman type space.For ϕ = U 0 f ∈ A 0,λL (D) we have then Once and for all we introduce all the operators to be considered.Fourier transforms on L 2 (R) and L 2 (T) play a very important role in this work, where T = S 1 is the unit circumference.We begin with the tensorial decomposition We use now polar coordinates for the first tensorial factor space.For z = (z 1 , ..., z n−1 ) ∈ C n−1 , we write z k = r k t k with r k ≥ 0 and t k ∈ T. For t = (t 1 , ..., t n−1 ) and r = (r 1 , ..., r n−1 ), we often write rt to mean z , and we identify z with (t, r).
Theorem 5.1 The unitary operator The poly-Bergman type space A 2 λL is isomorphic to the subspace Corollary 5.2 The restriction of W to the space A 2 λ(L) given by The last equation in (10) separates the variable y from the rest of variables; this means that certain independent solutions for it can be expressed in the form f (x, z )g(y) as shown below.But we must do the corresponding part for the first kind of equation in (10).In polar coordinates, the first kind of equation in (10) takes the form The general solution of the last equation in ( 11) is given by ψ 0 j n (t, ρ, x)y j n e −(sgn x)y/2 .
Since Ψ(t, ρ, x, y) has to be in L 2 (D, dη λ ), we must take only positive values of x.Morever, by rearranging polynomial terms we can express Ψ(t, ρ, x, y) as Let A 2,λL denote the space U 2 (A 2,λL ).In order to simplify our computations let's consider the function instead of the whole function Ψ given in (12).Then where Obviously Thus {d m j n } m∈Z n−1 , as in (13), belongs to A 2,λL if and only if Let R denote the left hand side of this equation for the particular case l k = 1, and let G(x, y) be the function χ + (x) λ j n (y).We have Thus, the function {d m j n } m∈Z n−1 = U 2 Ψ j n belongs to A 2,λL if and only if for each m and k = 1, ..., n − 1: Fixed m ∈ Z n−1 + , the general solution of this system of equations has the form where J = ( j 1 , ..., j n−1 ) and J = (J , j n ).Alternately, the general solution is given by For arbitrary m ∈ Z n−1 , the general solution of the system of differential equations (15) can also be written as where p k (ρ k ) is a polynomial of degree at most l k − 1 and whose coeficients are functions in x.Suppose that m = (m 1 , ..., m n−1 ) , and the set of solutions is reduced by the L 2 -condition.We have non-trivial solutions for L + m − e ≥ 0, they are given by Then the function U 2 Ψ j n belongs to A 2,λL if and only if Proof of Theorem 6.2.The image space Ã0,λL (D) = U 0 ( Ã2 λL (D n )) ⊂ L 2 (D, dη λ ) consists of all functions ϕ = U 0 f satisfying the equations (20) In polar coordinates, the first type equation in (20) takes the form Under the transformation Ψ = V 2 V 1 φ, the system of equations ( 20) is now equivalent to Thus the general solution of the this last equation has the form Ψ(t, ρ, x, y) = l n −1 ψ 0 j n (t, ρ, x)y j n e (sgn x)y/2 .