Polyharmonic Bergman spaces and Bargmann type transforms

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J. Math. Anal. Appl. ••• (••••) •••–•••

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Polyharmonic Bergman spaces and Bargmann type transforms Luís V. Pessoa ∗,1 , Ana Moura Santos ∗∗,1 Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 5 March 2016 Available online xxxx Submitted by T. Ransford

In the main result of the paper we prove the decomposition of polyharmonic Bergman spaces over the upper-half plane into spaces of polyanalytic functions. Then, we introduce the decomposition of polyharmonic Bergman spaces into the orthogonal sum of its true polyharmonic Bergman subspaces and we state isometric isomorphisms between the different true polyharmonic Bergman spaces. This allows us to define the k-th harmonic Hilbert component of a polyharmonic Bergman function and to prove closed formulas for the reproducing kernel functions of the true polyharmonic and the polyharmonic Bergman spaces. The harmonic complex Fourier transform is introduced in order to give an explicit description of the cartesian and the Laguerre harmonic components of the images of a Bargmann type transform for the true polyharmonic Bergman spaces. Finally, it is proved that the polyharmonic Bergman space of order j is isometric isomorphic to 2j copies of the corresponding Hardy space. © 2016 Elsevier Inc. All rights reserved.

Keywords: Beurling–Alhfors transform Bargmann type transform Harmonic components Kernel functions Polyharmonic Bergman space

1. Introduction Let j be a nonzero integer. A complex smooth function f defined on a domain U ⊂ C (non-empty, open and connected) and satisfying ∂zj f = 0 , j = 1, 2, . . .

and ∂z−j f = 0 , −j = 1, 2, . . . ,

respectively, is said to be a j-polyanalytic function on U . If j is a negative integer, then a j-polyanalytic function is also said to be a |j|-anti-polyanalytic function. Here, the iterated operators ∂zj and ∂zj are defined inductively by * Principal corresponding author. ** Corresponding author. E-mail addresses: [email protected] (L.V. Pessoa), [email protected] (A.M. Santos). Both authors were partially supported by Fundação para a Ciência e a Tecnologia through Centro de Análise Funcional, Estruturas Lineares e Aplicações of Instituto Superior Técnico, Universidade de Lisboa, Portugal (PEst-OE/MAT/UI4032/2011). 1

http://dx.doi.org/10.1016/j.jmaa.2016.12.034 0022-247X/© 2016 Elsevier Inc. All rights reserved.

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∂zj f := ∂zj−1 (∂z f )

and ∂zj f := ∂zj−1 (∂z f )

(j = 1, 2, . . .),

where 1 ∂z := 2



∂ ∂ +i ∂x ∂y



1 and ∂z := 2



∂ ∂ −i ∂x ∂y



denote the differential operators with respect to the conjugated complex variable z and to the complex variable z, respectively. The poly-Bergman space Aj2 (U ) consists of j-polyanalytic functions on U which 2 2 also belong to the Lebesgue space L2 (U, dA). Since f ∈ A−j (U ) if and only if f¯ ∈ Aj2 (U ), the space A−j (U ) 2 2 2 ¯ (U ). The Bergman space A (U ) and the anti-Bergman space A (U ) are will also here be denoted by A 1 −1 j ¯ 2 (U ), respectively. usually denoted by A 2 (U ) and A In recent studies we can find some connections of the theory of polyanalytic functions with wavelet theory (see, e.g., [1,17]), applications in Physics (see, e.g., [2,13]) and universality-like properties of rather general bi-analytic Bergman kernels [14]. Note that there are known notable differences between the analytic and the non-analytic polyanalytic functions. The structure of the zero set is a typical example of that. Indeed, in contrast to the analytic case, polyanalytic functions which are not analytic can admit non-isolated zeros within its domain. For example, if j = 2, . . . and ξk (k = 1, . . . , j − 1) are distinct unit vectors in the upper-half plane, then the following function f (z) := (ξ 1 z − zξ1 ) . . . (ξ j−1 z − zξj−1 ) is j-polyanalytic in the complex plain and z = 0 is a cluster point of the zero set of f . This set consists of 2j − 2 half-straight lines outgoing from the origin, precisely the half-straight lines outgoing from z = 0 and passing through ξk or through −ξk . We note that in the context of a wider scope of results, the set of non-isolated zeros of polyanalytic functions is described in [6]. A complex smooth function u over Π := {z ∈ C : Im z > 0} and satisfying Δj u = 0,

where Δ := 4∂z ∂z

and Δj u := Δj−1 (Δu)

is said to be a j-polyharmonic function on Π, for j = 1, 2, . . .. Hence, Δ denotes the Euclidean Laplacian given by the real derivatives ∂x2 + ∂y2 , where z := x + iy is in cartesian form. If u is a smooth function then Δj u = 4j ∂zj ∂zj u. Hence, it is evident that a j-polyanalytic function also is j-polyharmonic. For j = 1, 2, . . . the polyharmonic Bergman space Hj2 (Π) is defined as the subspace of the Lebesgue space L2 (Π, dA) consisting of j-polyharmonic functions, where dA := dxdy is the two-dimensional area Lebesgue measure. In this paper we abbreviate L2 (Π, dA) to L2 (Π), and whenever j = 1 we just write H 2 (Π) to denote the space of harmonic functions, which is more common in the literature (see, e.g. [5,12,25,30]). We know from [28, Theorem 2.8] that Hj2 (Π) is a closed subspace of the Lebesgue space L2 (Π) and, for every n, m = 0, 1, . . . and every z ∈ Π, it holds |∂zn ∂zm u(z)| ≤

M u , u ∈ Hj2 (Π) y n+m+1

(1)

where z := x + iy is in cartesian form and M is a positive constant only depending on n, m and j. Therefore, we can refer to Hj2 (Π) as a reproducing kernel Hilbert space (RKHS) of functions. Moreover, since Hj2 (Π) is closed in L2 (Π), we know that the polyharmonic Bergman projection QΠ,j is well defined as the orthogonal projection from L2 (Π) onto Hj2 (Π). As in the polyanalytic setting, we observe that several properties of polyharmonic functions play an important role in mathematical applications, for instance in physics and mechanics (see, e.g. [11,18,21, 23] and the references given there). Classical problems from elasticity theory and the Boggio–Hadamard

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conjecture for a clamped plate can be given as good examples of that. More recently, applications for polyanalytic functions have appeared in contexts of signal processing (see, e.g. [3,4]) and quantum physics (see, e.g. [13,33,39]). However, in the mathematical literature the study of the properties of functions spaces of polyanalytic and polyharmonic functions endowed with a Bergman type norm and their operators is still in an early stage. The 2-polyharmonic type functions, the so-called biharmonic type functions, have been studied with a particular interest. One can find several studies about the sign of the Green function for some weighted biharmonic Laplacians with Dirichlet boundary conditions (see, e.g. [10,15] and the references given there). There are also several papers, where the Bergman type norms of polyharmonic functions have been estimated by means of weighted Bergman norms of its derivatives (see, e.g. [16,25]). In others studies, the polyharmonic Bergman space over the unit disk has been decomposed onto the sum of weighted harmonic Bergman spaces (see, e.g. [24]), based on the classical Almansi representation theorem (see [31] for an unified approach). In [8] the authors make use of a decomposition of the polyharmonic Bergman space, called there cellular decomposition, which allows them to study the uniqueness of the polyharmonic Dirichlet problem in the weighted polyharmonic Bergman space over the unit disk. Some decompositions involving polyharmonic functions appeared in the mathematical literature in different contexts. For example, in [7] it as been investigated a decomposition of a function in the L2 Lebesgue space of the unit disk into the sum of functions in the polyharmonic Bergman space and in the image of the k-th power of the Laplacian of an integral operator with kernel given by higher order Green functions. The structure of function spaces of polyanalytic and polyharmonic functions, and their operators, have recently been described partially based on some special properties of the two-sided compression of the Beurling–Ahlfors transform, which is a particular case of a Calderon–Zygmund operator (see e.g. [19,26, 28,29,36,37]). The decomposition of poly-Bergman spaces into true poly-Bergman spaces (see (14)), which were introduced by N. Vasilevski in [34,35], and the validity of the Dzhuraev formulas (see (4)) allow us to introduce isometric isomorphisms between the different true polyharmonic Bergman spaces. Remark that this kind of assertions are of special interest since the well known theory of two-dimensional singular integral operators can be used in order to facilitate a more easy understanding of the relations in play. In the present paper, we follow this kind of reasoning and show how to obtain several properties concerning the structure of the polyharmonic Bergman spaces over the half-spaces and its relation with the Hardy spaces and the Lebesgue spaces in one real variable. In section 2, we prove that the j-polyharmonic Bergman space splits into the orthogonal sum of the j-poly-Bergman and j-anti-poly-Bergman spaces (see Theorem 2.4), i.e.

Hj2 (Π) = Aj2 (Π) ⊕ A¯ j2 (Π) , j = 1, 2, . . . . This is the main result of the paper and we prove it based on some special properties of SΠ the two-sided compression of the Beurling–Ahlfors transform to L2 (Π). As a consequence, we show that the projections QΠ,j belong to the ∗-algebra generated by SΠ . We will also remark that the two-sided compression of the ∗ Riesz transforms (of even order) coincide with the iterations of SΠ or SΠ . Based on this we obtain different representations for QΠ,j in terms of singular integral operators. In section 3, we define the decomposition of the polyharmonic Bergman spaces into its true polyharmonic Bergman subspaces. This can be achieved because the Hilbert spaces Hj2 (Π) are well ordered sets by inclusion. It is established that the true polyharmonic Bergman space splits into the true poly-Bergman and the true anti-poly-Bergman spaces and we define isometric isomorphisms between different true polyharmonic Bergman spaces. It is shown (see Theorem 3.4) that the sum of the iterations of the (two-sided) compression of the Beurling–Ahlfors transforms to its adjoint, acts in the harmonic Bergman space as the composition of a multiplication operator with a differential operator. This allows us to write an isometric isomorphism in a closed form from j copies of the harmonic Bergman space onto the j-polyharmonic Bergman space.

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The k-th harmonic Hilbert component of a function in the polyharmonic Bergman space is defined and we establish non-singular integral formulas for determining it. In section 4, we introduce a Hilbert basis for the true polyharmonic and the polyharmonic Bergman spaces. In relation with the unit disk setting (see e.g. [20,28]), one could say that the elements of this basis are the rational Zernike functions or the half-space rational functions. We then give closed formulas for the reproducing kernel functions of the true polyharmonic and the polyharmonic Bergman spaces over Π. In section 5, we construct isometric isomorphisms from L2 (R) onto the true polyharmonic Bergman spaces. The harmonic complex Fourier transform is introduced in order to achieve an explicit description of the cartesian and the Laguerre harmonic components (see Theorem 5.7) of the images of the mentioned Bargmann type transform. This result will be achieved based on properties of SΠ , in particular based on the fact that the action of the singular integral operator SΠ on the Bergman space coincides with the composition of a multiplication operator with a differential operator. Finally, in section 6, we will make use of the classical Paley–Wiener theorem and the results of the previous section, to prove that the polyharmonic Bergman space of order j is isometrically isomorphic to 2j copies of the Hardy space. 2. The polyharmonic Bergman spaces 2 The poly-Bergman space splits in its true poly-Bergman spaces A(j) (U ), which, for nonzero integers j, are defined by (also see, e.g. [19,29,34,35] and [36, Chap. 3 and 4]) 2 2 A(±1) (U ) := A±1 (U )

2 2 and A(j) (U ) := Aj2 (U )  Aj−sgn j (U ) , j = ±1.

(2)

In the framework of the upper half space, we note that the unitary operator V : L2 (Π) → L2 (Π)

,

V f (z) = f (−z)

(3)

2 ¯ 2 (Π) and A ¯ 2 (Π), respectively. transforms Aj2 (Π) and A(j) (Π) onto A j (j) We now introduce the poly-Bergman projection BU,j and the true poly-Bergman projection BU,(j) , which 2 are defined as the orthogonal projection of L2 (U ) onto Aj2 (U ) and A(j) (U ), respectively. Moreover, the   orthogonal projections BU,−j and BU,(−j) are also denoted by BU,j and BU,(j) , respectively. Remark that

from (2) it follows straightforwardly that BU,(j) = BU,j − BU,j−1

U,(j) = B U,j − B U,j−1 and B

(j = 2, 3, . . .).

It is well known that the true and the poly-Bergman projections over Π belong to the algebra generated by two-dimensional singular integral operators. To make this statement precise we need to introduce some definitions. Let SU ∈ B (L2 (U )) be the singular integral operator (see, e.g. [22,32]) SU f (z) := −

1 π

 U

f (w) dA(w). (w − z)2

Here B (H ) denotes the C ∗ -algebra of all bounded operators on the Hilbert space H . The operator SU coincides with the two-sided compression of the (unitary) Beurling–Ahlfors transform S := SC to the Lebesgue space L2 (U ), i.e. SU := χU SχU I, where χU I represents the multiplication operator by the characteristic function χU of U .

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Π,j lie in the algebra of singular integral operators follows Let now j be a positive integer. That BΠ,j and B from the fact that the upper-half plane admits (exact) Dzhuraev’s formulas, i.e. the following representations of poly-Bergman projections on Π are valid [36, Theorem 3.7] (also see [26, Corollary 3.2]) ∗ j BΠ,j = I − (SΠ )j (SΠ )

Π,j = I − (S ∗ )j (SΠ )j . and B Π

(4)

Then, from (2) together with (4) it follows ∗ j−1 ∗ j ∗ j−1 BΠ,(j) = (SΠ )j−1 (SΠ ) − (SΠ )j (SΠ ) = (SΠ )j−1 BΠ (SΠ ) ,

Π (SΠ )j−1 . Π,(j) = (S ∗ )j−1 (SΠ )j−1 − (S ∗ )j (SΠ )j = (S ∗ )j−1 B B Π Π Π

(5)

From the definition of poly-Bergman spaces it is clear that BΠ,j+1 BΠ,j = BΠ,j

Π,j+1 B Π,j = B Π,j . and B

(6)

Moreover, it follows from results of N. Vasilevski (see [34, Theorem 4.5]) that  L2 (Π) =

∞  j=1

 2 A(j) (Π) ⊕



∞  j=1

 2 A¯ (j) (Π) .

(7)

In particular, the following useful equality holds Π,k = 0 ; j, k = 1, 2, . . . . BΠ,j B

(8)

It is said that a linear operator P on a Hilbert space H is a partial isometry if it is an isometry from the orthogonal space of its kernel onto its image. The initial and the final spaces of P are defined to be the orthogonal space of Ker P and Im P , respectively. Note that (4) states that each (SΠ )j is a partial isometry ¯ 2 (Π) and of A 2 (Π), respectively. Since, with initial and final spaces given by the orthogonal spaces of A j j j (SΠ ) is a partial isometry, for j = 1, 2, . . ., then it is said that SΠ is a power partial isometry. Moreover, ∗ since SΠ is also a power partial isometry it is natural to say that SΠ is a ∗-power partial isometry. Concerning the properties of partial isometries, we remark the following well known result. Proposition 2.1. Let H be a Hilbert space and let P ∈ B (H ). The following statements are equivalent: (i) P is a partial isometry; (ii) P ∗ is a partial isometry; (iii) P ∗ P is a projection. If P, Q ∈ B (H ) are partial isometries then the following assertions hold: (iv) the initial and final spaces of P are the images of the projections P ∗ P and P P ∗ , respectively; (v) P Q is a partial isometry if and only if the projections P ∗ P and QQ∗ commute. For the reader convenience we decide to give a transparent proof of the next result, which is based in quite general properties of partial isometries. Theorem 2.2. [19,36] Let j = 1, 2, . . .. The operators 2 2 (SΠ )j : A(k) (Π) → A(k+j) (Π) , k = 1, 2, . . . ∗ j 2 2 (SΠ ) : A(k) (Π) → A(k−j) (Π) , −k = 1, 2, . . .

(9) (10)

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as well as the following ones ∗ j 2 2 (SΠ ) : A(k) (Π) → A(k−j) (Π) , j < k = 1, 2, . . .

(11)

2 2 (SΠ )j : A(k) (Π) → A(k+j) (Π) , j < −k = 1, 2, . . .

(12)

are isometric isomorphisms. Furthermore, ∗ j Ker(SΠ ) = Aj2 (Π)

and

¯ 2 (Π). Ker(SΠ )j = A j

(13)

Proof. Let j be a positive integer and let k be a nonzero integer. Define P := (SΠ )j BΠ,(k) . From the Π,j assertion (iii) of Proposition 2.1, to show that P is a partial isometry it is sufficient to note that B commute with BΠ,(k) , which follows from (6) and (8). Moreover, for k > 0 or j < −k one has ∗ j Π,j )BΠ,(k) = BΠ,(k) . P ∗ P = BΠ,(k) (SΠ ) (SΠ )j BΠ,(k) = BΠ,(k) (I − B 2 Hence, in any of the cases k > 0 and j < −k, the operator (SΠ )j is a unitary operator from A(k) (Π) onto the final space of P . Let now k be a positive integer. Considering (5), a simple manipulation shows that ∗ j ∗ j+k−1 P P ∗ = (SΠ )j BΠ,(k) (SΠ ) = (SΠ )j+k−1 BΠ (SΠ ) = BΠ,(j+k) . 2 Hence, the final space of P coincides with A(j+k) (Π) and (9) follows easily. Now assume that j < −k. From (4) together with (5) and (8), we obtain ∗ j ∗ −k−1  ∗ j P P ∗ = (SΠ )j BΠ,(k) (SΠ ) = (SΠ )j (SΠ ) BΠ (SΠ )−k−1 (SΠ ) ∗ −k−j−1  = (I − BΠ,j )(SΠ ) BΠ (SΠ )−k−j−1 (I − BΠ,j )

= (I − BΠ,j )BΠ,(k+j) (I − BΠ,j ) = BΠ,(k+j) . 2 Therefore, also in this case, the final space of P coincides with A(k+j) (Π) and (12) easily follows. To prove that (10) and (11) follow from (9) together with (12), we note that the unitary operator V defined in (3) can be used within the following equations ∗ j V (SΠ )j V = (SΠ )

,

2 2 V (A(j) (Π)) = A(−j) (Π).

From here, it is clear that (10) and (11) follow from (9) and (12), respectively. Finally, we remark that the left and the right-hand side of (13) are a straightforward consequence of the Dzhuraev’s formulas on the left and on the right-hand side of (4), respectively. 2 For a positive integer j, the following decompositions are evident from the definition of true poly-Bergman spaces

Aj2 (Π) =

j  k=1

2 A(k) (Π)

j ¯ 2 (Π). ¯ 2 (Π) =  A and A j (k)

(14)

k=1

In view of this, the isometric isomorphisms given in the previous result can be merged into new isometric isomorphisms acting on the poly-Bergman spaces. The next result can be considered as a consequence of Theorem 2.2.

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Theorem 2.3. Let j = 1, 2, . . .. The operators 2 (SΠ )j : Ak2 (Π) → Ak+j (Π)  Aj2 (Π) , k = 1, 2, . . . ∗ j 2 2 ) : Ak2 (Π) → Ak−j (Π)  A−j (Π) , −k = 1, 2, . . . (SΠ

as well as the following ones ∗ j 2 (SΠ ) : Ak2 (Π)  Aj2 (Π) → Ak−j (Π) , j < k = 1, 2, . . . 2 2 (Π) → Ak+j (Π) , j < −k = 1, 2, . . . (SΠ )j : Ak2 (Π)  A−j

are isometric isomorphisms. Furthermore, ∗ j Ker(SΠ ) = Aj2 (Π)

and

¯ 2 (Π) . Ker(SΠ )j = A j

Theorem 2.4. Let j = 1, 2 . . .. The following direct sum holds

Hj2 (Π) = Aj2 (Π) ⊕ A¯ j2 (Π).

(15)

Proof. Let u ∈ Hj2 (Π). Considering the equality in (8), it is sufficient to prove that u coincides with the ¯ 2 (Π). Let f be defined by sum of functions from Aj2 (Π) with functions from A j ∗ j f := (SΠ )2j (SΠ ) u.

If U ⊂ C is any domain and ψ ∈ L2 (U ) is a smooth function on U , then [38, see e.g. p. 61] the functions SU ψ and SU∗ ψ are also smooth on U and ∂z SU ψ(z) = ∂z ψ(z)

, ∂z SU∗ ψ(z) = ∂z ψ(z)

(z ∈ U ).

It is clear that f ∈ L2 (Π) and from the equalities ∗ j ∗ j ∗ j ∂z2j f = ∂z2j (SΠ )2j (SΠ ) u = ∂z2j (SΠ ) u = ∂zj ∂zj (SΠ ) u = ∂zj ∂zj u = 0,

we obtain that ∗ j 2 f = (SΠ )2j (SΠ ) u ∈ A2j (Π).

Therefore, there exist functions g and h such that f =g+h,

2 where g ∈ A2j (Π)  Aj2 (Π)

and h ∈ Aj2 (Π).

From Theorem 2.3 it follows that ∗ j ∗ j (SΠ ) f = (SΠ ) g ∈ Aj2 (Π).

(16)

∗ j ∗ j ∗ j (SΠ ) f = (SΠ ) (SΠ )2j (SΠ ) u Π,j )(I − BΠ,j )u = (I − B

(17)

On the other hand, one has

Π,j )u. = (I − BΠ,j − B

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Hence, from (16) and (17) we obtain that  ∗ j  ¯ 2 (Π). Π,j u ∈ A 2 (Π) + A u = (SΠ ) f + BΠ,j u + B j j

2

From the next result it follows that the projections QΠ,j also belong to the ∗-algebra generated by two-dimensional singular integral operators. Theorem 2.5. Let j = 1, 2 . . .. Then QΠ,j can be written as Π,j = 2I − (SΠ )j (S ∗ )j − (S ∗ )j (SΠ )j . QΠ,j = BΠ,j + B Π Π Different representations for QΠ,j can be given. Indeed, let j be a nonzero integer and let Sj ∈ B (L2 (C)) be the Riesz transform of even order −2j, i.e. (−1)j |j| Sj f (z) := π



(w − z)j−1 f (w)dA(w) (w − z)j+1

C

(z ∈ C),

together with the two-sided compression of Sj to L2 (Π) given by SΠ,j := χΠ Sj χΠ I. For j a positive integer we know from [26] that ∗ j SΠ,j = (SΠ )

and SΠ,−j = (SΠ )j .

Hence the next result follows straightforward. Corollary 2.6. Let j = 1, 2 . . .. Then QΠ,j can be written as QΠ,j = 2I − SΠ,−j SΠ,j − SΠ,j SΠ,−j . 3. Unitary operators on polyharmonic Bergman spaces In this section we use some special properties of the Beurling–Ahlfors transform to describe the structure of the true polyharmonic Bergman spaces. Since the polyharmonic Bergman spaces are well ordered by inclusion, one can define the true polyharmonic Bergman spaces similarly to the true poly-Bergman spaces 2 A(j) (Π), in the following way 2 H(1) (Π) := H 2 (Π)

2 2 and H(j) (Π) := Hj2 (Π)  Hj−1 (Π) , j = 2, 3, . . . .

It is clear that the polyharmonic Bergman spaces splits into the true polyharmonic Bergman spaces, in the following sense

Hj2 (Π) =

j 

2 H(k) (Π) , j = 1, 2, . . . .

(18)

k=1

The true polyharmonic Bergman projection QΠ,(j) is defined as the orthogonal projection of L2 (Π) 2 onto H(j) (Π). Hence, due to Theorem 2.4 we obtain the following statement.

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Theorem 3.1. Let j = 1, 2 . . .. The following direct sum holds 2 2 ¯ 2 (Π). H(j) (Π) = A(j) (Π) ⊕ A (j)

Furthermore, Π,(j) . QΠ,(j) = BΠ,(j) + B In the next result we state isomorphisms between the natural different pieces of polyharmonic Bergman spaces. Theorem 3.2. Let j, k = 1, 2 . . .. If 0 < k ≤ j, then the following operator is an isometric isomorphism ∗ j 2 2 (SΠ )j + (SΠ ) : H(k) (Π) → H(j+k) (Π).

Proof. It is evident that (13) can be rewritten in the following form ∗ j Π,k (S ∗ )j , k = 1, . . . , j. Π,k = B 0 = (SΠ ) BΠ,k = BΠ,k (SΠ )j = (SΠ )j B Π

(19)

Let us define the following auxiliary operator   ∗ j P := (SΠ )j + (SΠ ) QΠ,(k) , to easily obtain from (5) and (19) together with Theorem 3.1 that ∗ j ∗ j P P ∗ = ((SΠ )j + (SΠ ) )QΠ,(k) ((SΠ )j + (SΠ ) ) ∗ j Π,(k) )((SΠ )j + (S ∗ )j ) ) )(BΠ,(k) + B = ((SΠ )j + (SΠ Π ∗ j ∗ j  ) + (SΠ ) BΠ,(k) (SΠ )j = (SΠ )j BΠ,(k) (SΠ

(20)

Π,(j+k) = BΠ,(j+k) + B = QΠ,(j+k) . 2 Therefore, P is a partial isometry with final space given by H(j+k) (Π). Moreover, considering (8), (9) and (10) from Theorem 2.2, Theorem 2.5 and (19), we obtain ∗ j 2 P ∗ P = QΠ,(k) ((SΠ )j + (SΠ ) ) QΠ,(k) ∗ 2j = QΠ,(k) ((SΠ )2j + (SΠ ) − QΠ,j + 2I)QΠ,(k) ∗ 2j = QΠ,(k) ((SΠ )2j + (SΠ ) )QΠ,(k) + QΠ,(k)

Π,(k) (SΠ )2j BΠ,(k) + BΠ,(k) (S ∗ )2j B Π,(k) + QΠ,(k) =B Π = QΠ,(k) . Thus, the result follows straightforwardly. 2 The next result states that every iteration of the singular integral operator SΠ acting on the Bergman space coincides with the composition of a multiplication operator by a differential operator. This composition was considered in [1] to establish an isometric isomorphism between the Bergman and the true-poly-Bergman spaces.

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Theorem 3.3. [29, Theorem 3.3] Let j = 1, 2 . . .. Then

∂zj (z − z)j ϕ(z) , ϕ ∈ A 2 (Π), (SΠ ) ϕ(z) = j!

∂zj (z − z)j φ(z) ∗ j ¯ 2 (Π). , φ∈A (SΠ ) φ(z) = j! j

∗ j Based on Theorem 3.3, we show in the proof of the next result that the operator (SΠ )j + (SΠ ) acts in the harmonic Bergman space as the composition of a multiplication operator with a differential operator.

Theorem 3.4. Let j = 0, 1, . . .. Then the following operator R(j) : H (Π) → H 2

2 (j+1) (Π)

,



Δj y 2j ν(z) R(j) ν(z) = (2j)!

is an isometric isomorphism, where z := x + iy is in cartesian form. Proof. Let j = 1, 2, . . . and let ν ∈ H 2 (Π). From Theorem 2.4 it follows that there exist ϕ ∈ A 2 (Π) and ¯ 2 (Π) such that ν = ϕ + φ. Hence, considering (13) and Theorem 3.3 we obtain that φ∈A 

 ∗ j ∗ j ) ν(z) = (SΠ )j ϕ(z) + (SΠ ) φ(z) (SΠ )j + (SΠ



∂zj (z − z)j ϕ(z) ∂zj (z − z)j φ(z) = + j! j!



j 2j j Δ (z − z)2j φ(z) Δ (z − z) ϕ(z) + = 4j (2j)! 4j (2j)!

Δj (z − z)2j ν(z) = 4j (2j)!

2j j j Δ y ν(z) . = (−1) (2j)!

(21)

Based on Theorem 3.2, the proof is now easily completed. 2 Let j be a positive integer. We introduce first some notation for the proof of the result, which states that the j-polyharmonic Bergman space is a copy of j harmonic Bergman spaces. Let H be a Hilbert space and

let H j denote the Hilbert space of all j × 1 matrices (ak )jk=1 with entries in H . Theorem 3.5. Let j = 1, 2, . . .. Then the operator

Rj : H 2 (Π) j → Hj2 (Π) given by

Rj (νk )jk=1 (z)

:=

j  k=1



j−1  Δk y 2k νk+1 (z) , R(k−1) νk (z) = (2k)! k=0

is an isometric isomorphism, where z := x + iy is in cartesian form.

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11

2

Proof. The proof easily follows from (18) together with Theorem 3.4.

Let j be a positive integer. We observe that the construction of an isometric isomorphism between j copies of the Bergman space and the poly-Bergman space Aj2 (Π) easily follows from the standard theory of Hilbert spaces and from results e.g. in [1,19,29,36]. Theorem 3.6. Let j = 1, 2, . . . and k = 0, . . . , j − 1 and let u ∈ Hj2 (Π). Then there exist unique harmonic functions νk+1 in H 2 (Π) such that

j−1  Δk y 2k νk+1 (z) . u(z) = (2k)! k=0

Moreover, if z := x + iy is in cartesian form, then 2 H(k+1) (Π) = Δk [y 2k H 2 (Π)]

and

Hj2 (Π) =

j−1 

Δk [y 2k H 2 (Π)].

k=0

Proof. Just combine (18) and Theorems 3.4 and 3.5. 2 Let j be a positive integer and let u be a function in the polyharmonic Bergman space Hj2 (Π). The functions νk ∈ H 2 (Π) (k = 1, . . . , j) are said to be the harmonic Hilbert components of u if the following equality holds

j−1  Δk y 2k νk+1 (z) u(z) = (2k)!

i.e.

Rj (νk )jk=1 = u.

k=0

In this case, νk is said to be the k-th harmonic Hilbert component of u. Theorem 3.7. Let j = 1, 2 . . . and let u ∈ Hj2 (Π). For k = 1, . . . , j the k-th harmonic Hilbert component of u is given by νk (z) =

2 π

 Π

(z − w)k−2 (z − w)k−1 u(w)dA(w). Re (k − 1) − k (z − w)k (z − w)k+1

Proof. From Theorem 3.4 we know that there exists a unique function νk in the harmonic Bergman space that verifies the following equality R(k−1) νk = QΠ,(k) u , k = 1, . . . , j.

(22)

If, for k = 1, . . . , j the function νk is the solution of (22), then we obtain Rj (νk )jk=1 =

j  k=1

R(k−1) νk =

j 

QΠ,(k) u = u.

k=1

Therefore, the unique solution νk of (22) is the k-th harmonic Hilbert component of u ∈ Hj2 (Π). Let us define the following functions ν1 := QΠ u

∗ k and νk+1 := (−1)k QΠ ((SΠ )k + (SΠ ) )u , k = 1, . . . , j − 1.

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Then, for k = 2, . . . , j it easily follows from (20) and (21) that ∗ k−1 R(k−1) νk = (−1)k−1 R(k−1) QΠ ((SΠ )k−1 + (SΠ ) )u ∗ k−1 ∗ k−1 = ((SΠ )k−1 + (SΠ ) )QΠ ((SΠ )k−1 + (SΠ ) )u

= QΠ,(k) u. Hence, for k = 1, . . . , j the function νk is the k-th harmonic Hilbert component of u. Moreover, for k = 2, . . . , j from (19) we know that ∗ k−1 νk := (−1)k−1 QΠ ((SΠ )k−1 + (SΠ ) )u ∗ k−1 Π )((SΠ )k−1 + (SΠ ) )u = (−1)k−1 (BΠ + B

(23)

∗ k−1 Π (SΠ )k−1 u. ) u + (−1)k−1 B = (−1)k−1 BΠ (SΠ

Remark that the Bergman space A 2 (Π) is a reproducing kernel Hilbert space of functions (RKHS) with reproducing kernel function given by KΠ (z, w) = −

1 1 . π (z − w)2

(24)

Let Kz (w) := KΠ (w, z). Then, from the reproducing property, we know that     ∗ k−1 ∗ k−1 BΠ (SΠ ) u(z) = BΠ (SΠ ) u, Kz = u, (SΠ )k−1 Kz , z ∈ Π.

(25)

 Π (w, z) be the Let C denote the anti-linear isomorphism of complex conjugation acting on L2(Π), let K 2 ¯   reproducing kernel function for the space A (Π) and let Kz (w) := KΠ (w, z). Then     Π (SΠ )k−1 u(z) = u, (S ∗ )k−1 K  z = u, C(SΠ )k−1 Kz , z ∈ Π. B Π

(26)

Hence, from (23), (25) together with (26) it follows that   νk (z) = (−1)k−1 u, 2 Re(SΠ )k−1 Kz , z ∈ Π

(k = 1, . . . , j).

For k = 1, . . . , j due to Theorem 3.3, we then obtain

k−1 ∂w (w − w)k−1 π(k − 1)! (z − w)2

(w − z)k−1 1 (w − z)k−2 . k =− + (k − 1) π (z − w)k+1 (z − w)k

(SΠ )k−1 Kz (w) = −

2

4. More on the structure of polyharmonic Bergman spaces In what follows, for a nonzero integer j and a positive integer k, we consider (see [27, Sect. 9]) the functions ψj,k defined by √

(z − z)j−1 (z − i)k−1 2i k ; j, k = 1, 2, . . . ψj,k (z) := √ ∂zj−1 (z + i)k+1 π(j − 1)!

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13

together with the following relation ψj,k (z) = ψ −j,k (z) ; ±j, k = 1, 2, . . . . The functions ψj,k have similar properties to the Zernike polynomials in the unit disk. We can then say that they are the rational Zernike functions or the half-space rational functions. For positive integers n, m and j, we note that the functions ψn,m are the same as those in [1, Proposition 6], in which result it is established an Hilbert basis for the space Aj2 (Π). Proposition 4.1. [27, Proposition 9.1] Let j be a nonzero integer. Then {ψn,m : n = j}

and

{ψn,m : n sgn(j) = 1, . . . , j sgn(j)}

2 are Hilbert bases for the spaces A(j) (Π) and Aj2 (Π), respectively.

The next result follows from the equality on (7), stated by N. Vasilevski on the paper [34], Theorems 2.4 and 3.1 together with Proposition 4.1. Theorem 4.2. The following decomposition holds L2 (Π) =

+∞  j=1

2 H(j) (Π).

Furthermore, for j = 1, 2, . . . the following sets {ψn,m }

,

{ψn,m : n = ±j}

and

{ψn,m : ±n = 1, . . . , j}

2 are Hilbert bases for L2 (Π), H(j) (Π) and Hj2 (Π), respectively.

It is well known that poly-Bergman spaces Aj2 (Π) are reproducing kernel Hilbert spaces of functions. Explicit representations for the reproducing kernel functions KΠ,j (z, w) and KΠ,(j) (z, w) of the Hilbert 2 spaces Aj2 (Π) and A(j) (Π), respectively, are known (see [26,27,34]). As far as we know, they have appeared for the first time in [34], although not with the simplest formulation. More clear and computational formulas were given in [26,27] and will be partially discussed below. From (1) we know that the polyharmonic Bergman spaces Hj2 (Π) are RKHS of functions. Since the 2 spaces H(j) (Π) are closed subspaces of Hj2 (Π), then the true polyharmonic Bergman spaces are also RKHS of h h functions over Π. We denote by KΠ,j (z, w) and KΠ,(j) (z, w) the reproducing kernel functions for Hj2 (Π) and 2 h h h H(j) (Π), respectively. The kernel functions KΠ,1 (z, w) and KΠ,−1 (z, w) are also here denoted by KΠ (z, w) h

h and K Π (z, w), respectively. The reproducing kernel function for the harmonic Bergman space KΠ (z, w) is known for sometime (see, e.g. [5, Theorem 8.24]). Based on our previous arguments, one obtains from Theorem 3.1 and (24) that

h KΠ (z, w) = KΠ (z, w) + K Π (z, w) =

2 |z − w|2 − 2(x − t)2 . π |z − w|4

Here and henceforward, the equalities z = x + iy and w = t + is always refer to the Cartesian form of z and w, respectively.

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L.V. Pessoa, A.M. Santos / J. Math. Anal. Appl. ••• (••••) •••–•••

Theorem 4.3. Let j = 0, 1, . . .. Then

y 2j s2j KΠ (z, w) , KΠ,(j+1) (z, w) = (2j)! (2j)! 2j

y s2j j j K Π (z, w) , KΠ,(−j−1) (z, w) = Δz Δw (2j)! (2j)! Δjz Δjw

(27) (28)

where z = x + iy and w = t + is. Furthermore, h KΠ,(j+1) (z, w)

=

Δjz Δjw

y 2j s2j h K (z, w) . (2j)! (2j)! Π

Proof. From [27, Theorem 9.3] we know that

j (z − z)j (w − w)j ∂zj ∂w . KΠ,(j+1) (z, w) = − π(j!)2 (z − w)2 Considering (24) and that Δ = 4∂z ∂z , it then follows KΠ,(j+1) (z, w) = Δjz Δjw

(z − z)2j (w − w)2j K (z, w) . Π (2j)!4j (2j)!4j

(29)

Since, for every nonzero integer k, the following equality is evident KΠ,(−k) (z, w) = K Π,(k) (z, w), then from (29) we obtain (27) and (28). Finally, Theorem 3.1 implies that h KΠ,(j+1) (z, w) = KΠ,(j+1) (z, w) + K Π,(j+1) (z, w) 2j

 y s2j  j j = Δz Δ w KΠ (z, w) + K Π (z, w) (2j)! (2j)! 2j

s2j y j j h K (z, w) . 2 = Δz Δ w (2j)! (2j)! Π

Theorem 4.4. Let j = 1, 2, . . .. Then h KΠ,j (z, w)

   j 2(k−1)

 j + k − 1 |z − w| 2j 2j j−k j = − Re (w − z) (−1) . 2(j+k) π k j |z − w| k=1

Proof. From Theorem 3.1 it follows straightforwardly that h KΠ,j (z, w) = KΠ,j (z, w) + KΠ,−j (z, w)

= KΠ,j (z, w) + K Π,j (z, w) = 2 Re KΠ,j (z, w). We also know from [26, Corollary 2.5] that j KΠ,j (z, w) = − π

j

 j+k−1

j−k j k=1 (−1) k

j

|z − w|2(j−k) |z − w|2(k−1)

(z − w)2j

.

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15

Hence KΠ,j (z, w) = −

   j j j j + k − 1 (z − w)j−k 2(k−1) (−1)j−k |z − w| π (z − w)j+k k j k=1

   j  j j + k − 1 |z − w|2(k−1) j = − (z − w)2j (−1)j−k . k j π |z − w|2(j+k) k=1

The proof can be now easily completed. 2 The next result follows straightforwardly from (18) and Theorem 4.3. Theorem 4.5. Let j = 1, 2, . . . and let z := x + iy and w := t + is. Then

h KΠ,j (z, w) =

j−1 

Δkz Δkw

k=0

y 2k s2k h KΠ (z, w) . (2k)! (2k)!

Moreover,

KΠ,j (z, w) =

j−1 



y 2k s2k KΠ (z, w) , (2k)! (2k)!



y 2k s2k K Π (z, w) . (2k)! (2k)!

Δkz Δkw

k=0

KΠ,−j (z, w) =

j−1 

Δkz Δkw

k=0

5. Analogous Bargmann transforms Let Λ ⊂ R be a one-dimensional Lebesgue measurable set. By L2 (Λ) we will denote the Lebesgue space L (Λ, dt), where dt is the one-dimensional length Lebesgue measure. The proof that the operator in (30) is an isometric isomorphism can be found in [34, Theorem 2.4]. This result is the first Paley–Wienner theorem for the Bergman space that we are aware of. 2

Theorem 5.1. [29, Theorem 2.2] (See, also [34, Theorem 2.4].) The following operators are isometric isomorphisms

R : L2 (R+ , dt) → A 2 (Π)

,

1 Ra(z) = √ π

+∞  √

ta(t)eizt dt,

(30)

ta(t)e−izt dt.

(31)

0

¯ 2 (Π)  : L2 (R+ , dt) → A R

,

1  Ra(z) =√ π

+∞  √ 0

The weighted Bergman space A 2 (Π, dAλ ) is the function space of those analytic functions on Π, which also belong to the Lebesgue space L2 (Π, dAλ ) endowed with the measure dAλ (z) := y λ dA(z), where z := x + iy. For λ > −1, it is well known that A 2 (Π, dAλ ) is a Hilbert space endowed with the inner product induced by that of L2 (Π, dAλ ). The next result is said to be a Paley–Wiener type theorem for A 2 (Π, dAλ ).

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L.V. Pessoa, A.M. Santos / J. Math. Anal. Appl. ••• (••••) •••–•••

Theorem 5.2. [9, Theorem 1] For λ > −1, the complex Fourier transform

Fλc a(z)

=

2λ/2 πΓ(λ + 1)

+∞  a(t)eizt dt , z ∈ Π 0

defines an isometric isomorphism from L2 (R+ , dt/tλ+1 ) onto A 2 (Π, dAλ ). In [29] we showed that the next Paley–Wiener type theorem holds for the true poly-Bergman spaces. Theorem 5.3. [29, Theorem 3.4] Let j = 0, 1, . . .. The operators ∂zj [(z − z)j Ra(z)] j!

2 R(j) : L2 (R+ ) → A(j+1) (Π)

,

R(j) a(z) =

¯2 (j) : L2 (R+ ) → A R (j+1) (Π)

,

j j (j) a(z) = ∂z [(z − z) Ra(z)] R j!

are isometric isomorphisms. Furthermore, if y := (z − z)/(2i), then

R(j) a(z) =

j 

y k ϕk (z) =

k=0

(j) a(z) = R

j 

j 

Lk (y)φk (z)

k=0

y k ϕk (−z) =

k=0

j 

Lk (y)φk (−z),

k=0

where for k = 0, . . . , j the analytic cartesian components ϕk and the analytic Laguerre components φk satisfy the following conditions c c ϕk = F2k ak ∈ A 2 (Π, dA2k ) and φk = F2j bk ∈ A 2 (Π, dA2j ),

and the functions ak and bk are respectively given by   (2k)! √ 2k+1 j ak (t) := (−1) t a(t) , t ∈ R+ k k!   √  k j bk (t) := (−1) (2j)! t2k+1 (t − 1/2)j−k a(t) , k j+k

t ∈ R+ .

The Bargmann transform represents the classical isometric isomorphism from L2 (R) onto the Hilbert Fock space. In the next statement we introduce an analogous of the Bargmann transform for the harmonic Bergman space. Theorem 5.4. Let z := x + iy. Then, the following operator 1 R : L (R) → H (Π) , R a(z) = √ π h

2

is an isometric isomorphism.

2

h

+∞   −∞

|t|a(t)eixt e−y|t| dt

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17

Proof. From Theorems 3.1 and 5.1, it is clear that the mapping  2 (z) , M ((ak )2k=1 )(z) = Ra1 (z) + Ra

M : [L2 (R+ )]2 → H 2 (Π)

is an isometric isomorphism. Note that in matrix notation M is written as M = [R

 ]. R

Since the mapping T : L2 (R) → [L2 (R+ )]2

,

T a = (ak )2k=1

(32)

where the functions ak ∈ L2 (R+ ), for k = 1, 2, are defined by a1 (t) := a+ (t) := a(t) ,

t ∈ R+

a2 (t) := a− (t) := a(−t) ,

t ∈ R+

(33)

is also an isometric isomorphism, then M T is an isometric isomorphism from L2 (R) onto H 2 (Π). Moreover,  2 (z) M T a(z) = Ra1 (z) + Ra 1 =√ π

+∞  √

izt

ta(t)e

+∞  √

1 dt + √ π

0

1 =√ π

0

+∞ 

√

izt

ta(t)e

1 dt + √ π

0

1 =√ π

ta(−t)e−izt dt

+∞ 



0

√

−ta(t)eizt dt

(34)

−∞

|t|a(t)eixt e−y|t| dt

−∞

= Rh a(z), which completes the proof. 2 For λ > −1, we also define the harmonic complex Fourier transform

Fλh a(z)

2λ/2

= πΓ(λ + 1)

+∞  a(t)eixt e−y|t| dt , z ∈ Π −∞

where z := x + iy. In the next result we prove that the harmonic complex Fourier transform Fλh defines a bounded operator between the weighted Lebesgue space over R and the weighted harmonic Bergman space over Π. Proposition 5.5. The following map is a bounded operator Fλh : L2 (R, dt/|t|λ+1 ) → H 2 (Π, dAλ ).

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Proof. Similarly to (32) and (33), we define the map Tλ : L2 (R, dt/|t|λ+1 ) → [L2 (R+ , dt/tλ+1 )]2

,

Tλ a = (ak )2k=1 ,

where, by analogy with (33), the functions ak ∈ L2 (R+ , dt/tλ+1 ) are given by a1 (t) := a+ (t) := a(t) , t ∈ R+ a2 (t) := a− (t) := a(−t) , t ∈ R+ . It is clear that Tλ is an isometric isomorphism. We also consider the operator Vλ defined similarly to (3) by Vλ : L2 (Π, dAλ ) → L2 (Π, dAλ )

, Vλ f (z) = f (−z).

(35)

¯ 2 (Π, dAλ ). Moreover, in matrix notation one has It is evident that Vλ transforms A 2 (Π, dAλ ) onto A [ Fλc

Vλ Fλc ]Tλ a = Fλc a+ + Vλ Fλc a− .

Therefore, an analogous reasoning to the one given in (34), one obtains Fλc a+ + Vλ Fλc a− = Fλh a.

(36)

Together with Theorem 5.2, this completes the proof. 2 A complex smooth function u on a domain U and satisfying Δj u = 0 is said to be a j-polyharmonic function over U . Proposition 5.6. Let U ⊂ C be a simply connected domain, let j = 1, 2, . . ., and let u be a j-polyharmonic function on U . If Pk (z) are polynomials with degree k (k = 0, . . . , j − 1), then there exist unique harmonic functions νk and μk on U such that u(z) =

j−1 

(z − z) νk (z) = k

k=0

j−1 

Pk (z − z)μk (z).

(37)

k=0

Proof. Since U is a simply connected domain, due to [28, Proposition 2.1] we know that there exists a j-polyanalytic function f and a j-anti-polyanalytic function g such that u = f + g. We now proceed by induction in the variable j to prove the existence of analytic functions ϕk (k = 0, . . . , j − 1) such that f (z) =

j−1 

(z − z)k ϕk (z).

(38)

k=0

To achieve this it is sufficient that U be a domain. If j = 1, then the assertion is evident. Let j = 2, 3, . . .. The function ∂z f is (j − 1)-polyanalytic on U . Hence, by hypothesis, there exist analytic functions ψk such that ∂z f (z) =

j−2  k=0

Define the following j-polyanalytic function

(z − z)k ψk (z).

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h(z) := −

j−2  ψk (z) k=0

k+1

19

(z − z)k+1 .

It is clear that ∂z (f − h) = 0, i.e. f (z) = ϕ0 (z) + h(z), where ϕ0 (z) is analytic. Hence, for k = 0, . . . , j − 2, we define ϕk+1 (z) = −ψk (z)/(k + 1) to finish the proof of (38). Since g is j-anti-polyanalytic, then g is j-polyanalytic. From the above reasoning it follows the existence of analytic functions φk such that

g(z) =

j−1 

φk (z)(z − z)k =

k=0

j−1 

(−1)k φk (z)(z − z)k .

k=0

Therefore,

u(z) = f (z) + g(z) =

j−1 

νk (z)(z − z)k ,

where νk (z) = ϕk (z) + (−1)k φk (z).

k=0

We have proved the existence of harmonic functions νk (k = 0, . . . , j − 1) such that the decomposition on the left of (37) holds. To prove the uniqueness, it is sufficient to show that, if υk are harmonic on U and j−1 

υk (z)(z − z)k = 0,

k=0

then υk = 0, for k = 0, . . . , j − 1. As pointed out above there exist analytic functions ϕk and φk such that υk = ϕk + φk . Hence, it is straightforward that 0=

∂zj−1 ∂zj

j−1 

υk (z)(z − z)k = (−1)j−1 (j − 1)!ϕj−1 (z)

k=0

= ∂zj−1 ∂zj

j−1 

υk (z)(z − z)k = (j − 1)!φj−1 (z).

k=0

We conclude that υj−1 = 0 and by induction that υk = 0, for k = 0, . . . , j − 2. The existence and uniqueness of harmonic functions μk such that the equality on the right of (37) holds, is a straightforward consequence of the existence and uniqueness of constants cn,k and dn,k such that zn =

n 

cn,k Pk (z)

and Pn (z) =

k=0

n 

dn,k z k , n = 0, . . . , j − 1.

2

k=0

For a j-polyharmonic function u over a simply connected domain U ⊂ C, the functions νk and μk in the decomposition (37) are said to be the (k + 1)-th harmonic cartesian component and the (k + 1)-th harmonic Pk component of u, respectively. We focus on the case when Pk are the Laguerre polynomials

Lk (z) :=

k    k (−1)n n=0

n

n!

z n , k = 0, 1, . . . .

In Theorem 5.7 below we state that there is an isometry from L2 (R) onto each true polyharmonic Bergman space and furnish explicit representations for the harmonic cartesian components and the harmonic Laguerre components of its images.

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Theorem 5.7. Let j = 0, 1, . . . and let z := x + iy. The following operator h R(j)

: L (R) → H 2

2 (j+1) (Π)

,

h R(j) a(z)



Δj y 2j Rh a(z) = (2j)!

is an isometric isomorphism. Furthermore, h R(j) a(z)

=

j 

k

y νk (z) =

k=0

j 

Lk (y)μk (z),

k=0

where the harmonic cartesian components νk and the harmonic Laguerre components μk satisfy the following conditions h h νk = F2k ak ∈ A 2 (Π, dA2k ) and μk = F2j bk ∈ A 2 (Π, dA2j ),

and, for k = 0, . . . , j, the functions ak and bk are respectively given by   (2k)! k  j |t| |t|a(t) , t ∈ R ak (t) := (−1) k k!    (2j)! k j bk (t) := |t| (1 − 2|t|)j−k |t|a(t) , t ∈ R. j−k k 2 k

h Proof. That R(j) is an isometric isomorphism follows from Theorems 3.4 and 5.4. From the proof of Theorem 5.4 we know that

 −, Rh a = Ra+ + Ra

(39)

where the functions a± are defined by (33). Therefore, from (39) together with Theorems 5.1 and 5.3, we obtain h R(j) a(z) =

 − (z)] Δj [y 2j Ra+ (z) + y 2j Ra Δj [y 2j Rh a(z)] = (2j)! (2j)!

= (−1)j

 − (z)] ∂zj [(z − z)j Ra+ (z)] ∂ j [(z − z)j Ra + (−1)j z j! j!

(j) a− (z)). = (−1)j (R(j) a+ (z) + R Hence, Theorems 5.2 and 5.3 imply that h R(j) a(z) =

j 

− y k (ϕ+ k (z) + ϕk (−z))

k=0

=

j 

− Lk (y)(φ+ k (z) + φk (−z)),

k=0

where, for k = 0, . . . , j, we have ± c ± 2 c ± 2 ϕ± k = F2k ak ∈ A (Π, dA2k ) and φk = F2j bk ∈ A (Π, dA2j ).

(40)

± 2 + 2k+1 The functions a± ) and in L2 (R+ , dt/t2j+1 ), respectively, and are given by k and bk are in L (R , dt/t

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  (2k)! √ 2k+1 ± j := (−1) t a (t), k! k   √ j  ± (2j)! t2k+1 (1/2 − t)j−k a± (t). bk (t) := k

a± k (t)

k

For k = 0, . . . , j, due to (35) and (36) together with (40), it follows that − c + c − h ϕ+ k (z) + ϕk (−z) = (F2k ak + V2k F2k ak )(z) = F2k ak (z), − c + c − h φ+ k (z) + φk (−z) = (F2j bk + V2j F2j bk )(z) = F2j bk (z),

where the functions ak and bk are defined on the real line R by − ak (t) := χ+ (t)a+ k (t) + χ− (t)ak (−t)    (2k)! j |t|2k+1 a(t) , = (−1)k k k! − bk (t) := χ+ (t)b+ k (t) + χ− (t)bk (−t)    j  (2j)! |t|2k+1 (1/2 − |t|)j−k a(t) , = k

and χ+ and χ− are the characteristic functions of R+ and R− , respectively.

2

The next result follows straightforwardly from (18) and Theorem 5.7. Theorem 5.8. Let j = 1, 2, . . .. Then the following operator is an isometric isomorphism

Rjh : L2 (R) j → Hj2 (Π)

,

Rjh (ak )jk=1 (z) =

j−1 

h R(k) ak (z).

k=0

6. Isomorphism between copies of the Hardy space The Hardy space A∂2 (Π) is the Hilbert space of functions that consists of those analytic functions ϕ on Π which have finite norm given by  ϕ2∂

|ϕ(x + iy)|2 dx.

= sup y>0

R

It is well known that the complex Fourier transform F : L (R ) → A c

2

+

2 ∂ (Π)

,

1 F a(z) = √ 2π c

+∞  a(t)eizt dt 0

is an isometric isomorphism. A function ϕ ∈ A∂2 (Π) admits non-tangential limits almost at every point t ∈ R, i.e. the following function of real variable a(t) := lim ϕ(z) z→t

whenever Π z → t ∈ R

nontangentially

is well defined, for almost every t ∈ R, and a(t) ∈ L2 (R). The real Fourier transform F ∈ B (L2 (R)) defined in the following way

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1 F a(x) = √ 2π



a(t)e−ixt dt,

x∈R

R

acts isometrically from A∂2 (Π) onto L2 (R+ ), i.e. the following operator defines an onto unitary operator

F : A∂2 (Π) → L2 (R+ ). Thus, from Theorem 5.8 we can conclude the following result. Theorem 6.1. For j = 1, 2, . . . the operator

Wjh : A∂2 (Π) 2j → Hj2 (Π)

,

j h Wjh (ϕk )2j k=1 = Rj (ak )k=1

where ak (t) = χ+ F ϕ2k−1 (t) + χ− F ϕ2k (−t)

;

k = 1, . . . , j

is an isometric isomorphism. References [1] L.D. Abreu, Super-wavelets versus poly-Bergman spaces, Integral Equations Operator Theory 73 (2012) 177–193. [2] L.D. Abreu, P. Balazs, M. de Gosson, Z. Mouayn, Discrete coherent states for higher Landau levels, Ann. Phys. 363 (2015) 337–353. [3] L.D. Abreu, H.G. Feichtinger, Function spaces of polyanalytic functions, in: A. Vasiliev (Ed.), Harmonic and Complex Analysis and Its Applications, in: Trends Math., Birkhäuser, Basel, 2014, pp. 1–38. [4] L.D. Abreu, K. Gröchenig, Banach Gabor frames with Hermite functions: polyanalytic spaces from the Heisenberg group, Appl. Anal. 91 (2012) 1981–1997. [5] S. Axler, P. Bourdon, W. Ramey, Harmonic Function Theory, Springer-Verlag, New York, 1992. [6] M.B. Balk, Polyanalytic Functions, Akademie Verlag, Berlin, 1991. [7] H. Begehr, Orthogonal decompositions of the function space L2 (D; C), J. Reine Angew. Math. 549 (2002) 191–219. [8] A. Borichev, H. Hedenmalm, Weighted integrability of polyharmonic functions, Adv. Math. 264 (2014) 464–505. [9] P. Duren, E.A. Gallardo-Guitíerrez, A. Montes-Rodrígues, A Paley–Wiener theorem for Bergman spaces with application to invariant subspaces, Bull. Lond. Math. Soc. 39 (2007) 459–466. [10] M. Engliš, J. Peetre, Green functions and eigenfunction expansions for the square of the Laplace–Beltrami operator on plane domains, Ann. of Math. 181 (2002) 463–500. [11] F. Gazzola, H. Grunau, G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag, Berlin, Heidelberg, 2010. [12] K. Guo, D. Zheng, Toeplitz algebra and Hankel algebra on the harmonic Bergman space, J. Math. Anal. Appl. 276 (2002) 213–230. [13] A. Haimi, H. Hedenmalm, The polyanalytic Ginibre ensembles, J. Stat. Phys. 153 (2013) 10–47. [14] A. Haimi, H. Hedenmalm, Asymptotic expansion of polyanalytic Bergman kernels, J. Funct. Anal. 267 (2014) 4667–4731. [15] H. Hedenmalm, A computation of Green functions for the Weighted biharmonic operators Δ|z|2λ Δ, λ > −1, Duke Math. J. 75 (1994) 51–78. [16] Z. Hu, M. Pavlović, X. Zhang, The mixed norm spaces of polyharmonic functions, Potential Anal. 27 (2007) 167–182. [17] O. Hutník, Wavelets from Laguerre polynomials and Toeplitz-type operators, Integral Equations Operator Theory 71 (2011) 357–388. [18] A.I. Kalandiya, Mathematical Methods of Two-Dimensional Elasticity, MIR Publishers, Moscow, 1975. [19] Yu.I. Karlovich, L.V. Pessoa, C ∗ -algebras of Bergman type operators with piecewise continuous coefficients, Integral Equations Operator Theory 57 (2007) 521–565. [20] A.D. Koshelev, On the kernel function of the Hilbert space of functions polyanalytic in a disc, translation from Dokl. Akad. Nauk SSSR 232 (1977) 277–279. [21] A.D. Koshelev, An application of polyharmonic functions in the theory of elasticity, translation from Sov. Phys. Dokl. 22 (1977) 129–130. [22] S.G. Mikhlin, S. Prössdorf, Singular Integral Operators, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1986. [23] N.I. Muskhelishvili, Some Basic Problems of Mathematical Elasticity Theory, Nauka, Moscow, 1968. [24] M. Pavlović, Decompositions of Lp and Hardy spaces of polyharmonic functions, J. Math. Anal. Appl. 216 (1997) 499–509. [25] M. Pavlović, Hardy–Stein type characterization of harmonic Bergman spaces, Potential Anal. 32 (2010) 1–15. [26] L.V. Pessoa, The method of variation of the domain for poly-Bergman spaces, Math. Nachr. 17–18 (2013) 1850–1862. [27] L.V. Pessoa, Planar Beurling transform and Bergman type spaces, Complex Anal. Oper. Theory 8 (2014) 359–381; L.V. Pessoa, Complex Anal. Oper. Theory 9 (2015) 1245–1247 (Erratum).

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[28] L.V. Pessoa, On the structure of polyharmonic Bergman type spaces over the unit disk, Complex Var. Elliptic Equ. 60 (12) (2015) 1668–1684. [29] L.V. Pessoa, A. Moura Santos, Theorems of Paley–Wiener type for spaces of polyanalytic functions, in: V. Mityushev, M.V. Ruzhansky (Eds.), Current Trends in Analysis and Its Applications, Research Perspectives, Birkhäuser, Basel, 2015, pp. 605–613. [30] W.C. Ramey, H. Yi, Harmonic Bergman functions on half-spaces, Trans. Amer. Math. Soc. 348 (1996) 613–627. [31] G. Ren, H.R. Malonek, Decomposing kernels of iterated operators. A unified approach, Math. Methods Appl. Sci. 30 (2007) 1037–1047. [32] E.M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, New Jersey, 1993. [33] A. Torre, Generalized Zernike or disc polynomials: an application in quantum optics, J. Comput. Appl. Math. 222 (2008) 622–644. [34] N.L. Vasilevski, On the structure of Bergman and poly-Bergman spaces, Integral Equations Operator Theory 33 (1999) 471–488. [35] N.L. Vasilevski, Poly-Fock spaces, Oper. Theory Adv. Appl. 117 (2000) 371–386. [36] N.L. Vasilevski, Poly-Bergman spaces and two-dimensional singular integral operators, Oper. Theory Adv. Appl. 171 (2006) 349–359. [37] N.L. Vasilevski, Commutative Algebras of Toeplitz Operators on the Bergman Space, Oper. Theory Adv. Appl., vol. 185, Birkhäuser Verlag, 2008. [38] I.N. Vekua, Generalized Analytic Functions, Pergamon Press/Addison–Wesley, Oxford/Reading, Mass., 1962. [39] A. Wunsche, Generalized Zernike or disc polynomials, J. Comput. Appl. Math. 174 (2005) 135–163.