Poly-Bergman spaces and two-dimensional

Operator Theory: Advances and Applications, Vol. 1, 1–11 c 2005 Birkh¨ ° auser Verlag Basel/Switzerland

Poly-Bergman spaces and two-dimensional singular integral operators N. Vasilevski Abstract. We describe a direct and transparent connection between the polyBergman type spaces on the upper half-plane and certain two dimensional singular integral operators. Mathematics Subject Classification (2000). 30G30, 45P05, 47B38. Keywords. Poly-Bergman spaces, singular integral operators, poly-Bergman projections.

1. Introduction We show that there is a direct and transparent connection between the polyBergman type spaces and certain two dimensional singular integral operators. Recall that the poly-Bergman spaces A2n (Π) and Ae2n (Π) on the upper halfplane Π, of analytic and anti-analytic functions respectively, are defined as the subspaces of L2 (Π), endowed with the standard Lebesgue plane measure dv(z) = dxdy, z = x + iy, and consist of functions satisfying the following equations µ ¶n µ ¶n ∂ 1 ∂ ∂ ϕ= n +i ϕ = 0, n ∈ N, ∂z 2 ∂x ∂y and

µ

∂ ∂z

¶n ϕ=

1 2n

µ

∂ ∂ −i ∂x ∂y

¶n ϕ = 0,

n ∈ N,

respectively. We introduce as well the following singular integral operators bounded on L2 (Π): Z 1 ϕ(ζ) dv(ζ) (SΠ ϕ)(z) = − π Π (ζ − z)2 This work was partially supported by CONACYT Project U40654-F, M´ exico.

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N. Vasilevski

and its adjoint ∗ (SΠ ϕ)(z) = −

1 π

Z Π

ϕ(ζ) dv(ζ). (ζ − z)2

A. Dzhuraev [4, 5] showed that for a bounded domain D with smooth boundeD,n of L2 (D) onto the spaces A2n (D) ary the orthogonal projections BD,n and B and Ae2n (D), respectively, can be expressed in the form ∗ n BΠ,n = I − (SD )n (SD ) + Kn

and

∗ n eΠ,n = I − (SD en, B ) (SD )n + K

e n are compact operators. Recently J. Ram´ırez and I. Spitkovsky where Kn and K [8] proved that in the case of the upper half-plane Π the compact summands Kn e n in the above formulas are equal to zero. Using this result Yu. Karlovich and and K ∗ L. Pessoa [7] described the action of the operators SΠ and SΠ on the poly-Bergman spaces, obtaining the statements of Theorem 3.5 below. In the paper we propose another, more direct and transparent, approach to the problem, which follows the ideas of [9, 10] and gives the precise information ∗ . In Section 2 we present necessary facts from about the structure of SΠ and SΠ [9, 10]. The core result of the paper is contained in Theorems 3.1 and 3.2 and ∗ gives a simple (functional) model for the operators SΠ and SΠ : each of them is unitary equivalent to the direct sum of two unilateral shifts, forward and backward, both taken with the infinite multiplicity. This fact permits us an easy access to the majority of the properties of these operators. The most important properties, in the context of the paper, are given by the subsequent Theorems 3.5 and 3.7.

2. Poly-Bergman spaces Let Π be the upper half-plane in C, consider the space L2 (Π) endowed with the usual Lebesgue plane measure dv(z) = dxdy, z = x + iy. Denote by A2 (Π) its Bergman subspace, i.e., the subspace which consists of all functions analytic in Π. It is well known that the Bergman projection BΠ of L2 (Π) onto A2 (Π) has the form Z 1 ϕ(ζ) (BΠ ϕ)(z) = − dv(ζ). π Π (z − ζ)2 In addition to the Bergman space A2 (Π) introduce the space Ae2 (Π) as the subspace of L2 (Π) consisting of all functions anti-analytic in Π. Further, analogously to the Bergman spaces A2 (Π) and Ae2 (Π), introduce the spaces of poly-analytic and poly-anti-analytic functions (see, for example, [1, 2, 4, 5]), the poly-Bergman spaces. We define the space A2n (Π) of n-analytic functions as the subspace of L2 (Π) of all functions ϕ = ϕ(z, z) = ϕ(x, y), which satisfy the equation µ ¶n µ ¶n ∂ ∂ ∂ 1 +i ϕ= n ϕ = 0. ∂z 2 ∂x ∂y

Poly-Bergman spaces and two-dimensional singular integral operators

3

Similarly, we define the space Ae2n (Π) of n-anti-analytic functions as the subspace of L2 (Π) of all functions ϕ = ϕ(z, z) = ϕ(x, y), which satisfy the equation µ ¶n µ ¶n ∂ 1 ∂ ∂ ϕ= n −i ϕ = 0. ∂z 2 ∂x ∂y Of course, we have A21 (Π) = A2 (Π) and Ae21 (Π) = Ae2 (Π), for n = 1, as well as A2n (Π) ⊂ A2n+1 (Π) and Ae2n (Π) ⊂ Ae2n+1 (Π), for each n ∈ N. Finally introduce the space A2(n) (Π) of true-n-analytic functions by A2(n) (Π) = A2n (Π) ª A2n−1 (Π), for n > 1, and by A2(1) (Π) = A21 (Π); and, symmetrically, introduce the space Ae2(n) (Π) of true-n-anti-analytic functions by Ae2(n) (Π) = Ae2n (Π) ª Ae2n−1 (Π), for n > 1, and by Ae2(1) (Π) = Ae21 (Π), for n = 1. We have, of course, n n M M A2n (Π) = A2(k) (Π) and Ae2n (Π) = Ae2(k) (Π). k=1

k=1

To formulate the main result of this section we need more definitions. We start by introducing two unitary operators. Define the unitary operator U1 = F ⊗ I : L2 (Π) = L2 (R) ⊗ L2 (R+ ) −→ L2 (R) ⊗ L2 (R+ ), where the Fourier transform F : L2 (R) → L2 (R) is given by Z 1 e−ixξ f (ξ) dξ. (F f )(x) = √ 2π R The second unitary operator

(1)

(2)

U2 : L2 (Π) = L2 (R) ⊗ L2 (R+ ) −→ L2 (R) ⊗ L2 (R+ ) is given by (U2 ϕ)(x, y) = p

1 2|x|

ϕ(x,

y ). 2|x|

(3)

Then the inverse operator U2−1 = U2∗ : L2 (R) ⊗ L2 (R+ ) −→ L2 (R) ⊗ L2 (R+ ) acts as follows, p (U2−1 ϕ)(x, y) = 2|x| ϕ(x, 2|x| · y). Recall (see, for example, [3]), that the Laguerre polinomial Ln (y) of degree n, n = 0, 1, 2, ..., and type 0 is defined by ey dn −y n Ln (y) = L0n (y) = (e y ) n! dy n n X (−y)k n! , y ∈ R+ , (4) = k!(n − k)! k! k=0

4

N. Vasilevski

and that the system of functions `n (y) = e−y/2 Ln (y),

n = 0, 1, 2, ...

(5)

forms an orthonormal basis in the space L2 (R+ ). Denote by Ln , n = 0, 1, 2, ..., the one-dimensional subspace of L2 (R+ ) generated by the function `n (y). The main result of the section reads as follows. Theorem 2.1. The unitary operator U = U2 U1 : L2 (R) ⊗ L2 (R+ ) → L2 (R) ⊗ L2 (R+ ) provides the following isometrical isomorphisms of the above spaces: 1. Isomorphic images of poly-analytic spaces U

: A2(n) (Π) −→ L2 (R+ ) ⊗ Ln−1 ,

U

: A2n (Π) −→ L2 (R+ ) ⊗

n−1 M

Lk ,

k=0

U

:

∞ M

A2(k) (Π) −→ L2 (R+ ) ⊗ L2 (R+ ).

k=1

2. Isomorphic images of poly-anti-analytic spaces U

:

Ae2(n) (Π) −→ L2 (R− ) ⊗ Ln−1 ,

U

:

Ae2n (Π) −→ L2 (R− ) ⊗

n−1 M

Lk ,

k=0

U

:

∞ M

Ae2(k) (Π) −→ L2 (R− ) ⊗ L2 (R+ ).

k=1

3. Furthermore we have the following decomposition of the space L2 (Π) L2 (Π)

=

∞ M (A2(k) (Π) ⊕ Ae2(k) (Π)) k=1

=

∞ M k=1

A2(k) (Π) ⊕

∞ M

Ae2(k) (Π).

k=1

3. Two-dimensional singular integral operators We introduce the following singular integral operators bounded on L2 (Π): Z 1 ϕ(ζ) dv(ζ) (SΠ ϕ)(z) = − π Π (ζ − z)2

Poly-Bergman spaces and two-dimensional singular integral operators and its adjoint ∗ (SΠ ϕ)(z) = −

1 π

Z Π

5

ϕ(ζ) dv(ζ). (ζ − z)2

∗ SΠ

Note, that the operators SΠ and are the restrictions onto the upper halfplane Π of the following classical two-dimensional singular integral operators over C = R2 , Z Z 1 ϕ(ζ) ϕ(ζ) 1 ∗ (SR2 ϕ)(z) = − dv(ζ) and (SR2 ϕ)(z) = − dv(ζ), π R2 (ζ − z)2 π R2 (ζ − z)2 which are given in terms of the Fourier transform as follows, ζ SR2 = F −1 F ζ

and

−1 ζ SR∗ 2 = SR−1 F, 2 = F ζ

(6)

where ζ = ξ + iη = (ξ, η), and the Fourier transform F is given by Z 1 (F ϕ)(ζ) = e−iζ·z ϕ(z) dv(z), 2π R2 where z = x + iy = (x, y), and ζ · z = ξx + ηy. By (6) these operators admit the following representations: SΠ

= (I ⊗ χ+ I)SR2 (I ⊗ χ+ I) ξ − iη = (I ⊗ χ+ I)(F −1 ⊗ F −1 ) (F ⊗ F )(I ⊗ χ+ I) ξ + iη

(7)

and ∗ SΠ

= =

(I ⊗ χ+ I)SR∗ 2 (I ⊗ χ+ I) ξ + iη (I ⊗ χ+ I)(F −1 ⊗ F −1 ) (F ⊗ F )(I ⊗ χ+ I), ξ − iη

where ξ, η ∈ R, and the one-dimensional Fourier transform F is given by (2). Let us introduce the following integral operators Z y t − y2 (S+ f )(y) = −f (y) + e e 2 f (t) dt, 0 Z ∞ y t (S− f )(y) = −f (y) + e 2 e− 2 f (t) dt, y

which, as we will see later on, are bounded on L2 (R+ ) and are mutually adjoint. As in Section 2 we will use the unitary operator U = U2 U1 : L2 (R) ⊗ L2 (R+ ) −→ L2 (R) ⊗ L2 (R+ ), where the operators U1 and U2 are given by (1) and (3) respectively. Theorem 3.1. The unitary operator U = U2 U1 gives an isometrical isomorphism of the space L2 (Π) = [L2 (R+ ) ⊗ L2 (R+ )] ⊕ [L2 (R− ) ⊗ L2 (R+ )] under which the

6

N. Vasilevski

∗ two-dimensional singular integral operators SΠ and SΠ are unitary equivalent to the following operators

U SΠ U −1

=

(I ⊗ S+ ) ⊕ (I ⊗ S− ),

∗ U SΠ U −1

=

(I ⊗ S− ) ⊕ (I ⊗ S+ ).

Proof. By the representation (7) we have S1

U1 SΠ U1−1 = (F ⊗ I)SΠ (F −1 ⊗ I) ξ − iη (I ⊗ χ+ I)(I ⊗ F −1 ) (I ⊗ F )(I ⊗ χ+ I). ξ + iη

= =

The operator U2 is unitary on both L2 (R+ ) and L2 (R), and furthermore it commutes with χR+ I. Direct calculation shows that U2 (I ⊗ F −1 )

1 signx − iη ξ − iη (I ⊗ F )U2−1 = (I ⊗ F −1 ) 21 (I ⊗ F ). ξ + iη 2 signx + iη

Thus S2

=

U SΠ U −1 = U2 S1 U2−1 1

= (χ+ I ⊗ χ+ I)(I ⊗ F −1 ) 21 +

(χ− I ⊗ χ+ I)(I ⊗

=

∗ −1 U U SΠ

2 1 F −1 ) 21 2

− iη (I ⊗ F )(χ+ I ⊗ χ+ I) + iη + iη (I ⊗ F )(χ− I ⊗ χ+ I) − iη

and S2∗

1

= (χ+ I ⊗ χ+ I)(I ⊗ F −1 ) 21 +

(χ− I ⊗ χ+ I)(I ⊗

2 1 F −1 ) 21 2

+ iη (I ⊗ F )(χ+ I ⊗ χ+ I) − iη − iη (I ⊗ F )(χ− I ⊗ χ+ I). + iη

The symbols of the two convolution operators 1 − iη Se+ = F −1 21 F 2 + iη

and

1 + iη Se− = F −1 21 F, 2 − iη

which are obviously bounded on L2 (R), admit the following representations, 1 2 1 2

− iη = −1 − + iη

1 4

iη + + η2

1 4

1 2

+ η2

and

1 2 1 2

+ iη = −1 + − iη

1 4

iη + + η2

1 4

respectively. Using the formulas 17.23.14 and 17.23.15 of [6] we have ¶ r µ 1 ¶ r µ |y| π − |y| π iη = F 1 2 2 = e 2 , sign y e− 2 , F 1 2 2 2 4 +η 4 +η

1 2

+ η2

,

Poly-Bergman spaces and two-dimensional singular integral operators and thus (Se+ f )(y) = =

7

Z r 1 π − |t−y| −f (y) + √ e 2 (1 − sign(t − y))f (t)dt 2π R 2 Z |t−y| −f (t) + e− 2 χ− (t − y)f (t)dt R

and (Se− f )(y)

Z r 1 π − |t−y| = −f (y) + √ e 2 (1 + sign(t − y))f (t)dt 2π R 2 Z |t−y| = −f (t) + e− 2 χ+ (t − y)f (t)dt. R

Then the operators S+ = χ+ Se+ χ+ I|L2 (R+ ) and S− = χ+ Se− χ+ I|L2 (R+ ) , acting on L2 (R+ ), are as follows: Z |t−y| e− 2 χ− (t − y)f (t)dt (S+ f )(y) = −f (t) + R+

Z

y

= −f (y) + e− 2

y

t

e 2 f (t)dt 0

and

Z e−

(S− f )(y) = −f (t) +

|t−y| 2

χ+ (t − y)f (t)dt

R+

Z

∞

y

= −f (y) + e 2

t

e− 2 f (t)dt.

y

Thus finally U SΠ U −1

= (χ+ I ⊗ χ+ I)(I ⊗ Se+ )(χ+ I ⊗ χ+ I) + (χ− I ⊗ χ+ I)(I ⊗ Se− )(χ− I ⊗ χ+ I) = χ+ I ⊗ S+ + χ− I ⊗ S− = (I ⊗ S+ ) ⊕ (I ⊗ S− )

and ∗ −1 U SΠ U

= (χ+ I ⊗ χ+ I)(I ⊗ Se− )(χ+ I ⊗ χ+ I) + (χ− I ⊗ χ+ I)(I ⊗ Se+ )(χ− I ⊗ χ+ I) = χ+ I ⊗ S− + χ− I ⊗ S+ = (I ⊗ S− ) ⊕ (I ⊗ S+ ),

where the last lines in both representations are written according to the splitting L2 (Π) = [L2 (R+ ) ⊗ L2 (R+ )] ⊕ [L2 (R− ) ⊗ L2 (R+ )] . ¤

8

N. Vasilevski We continue to use an orthonormal basis `n (y) = e−y/2 Ln (y),

n = 0, 1, 2, ...,

of the space L2 (R+ ), where the Lagguerre polynomials Ln (y) are given by (4). Theorem 3.2. For each admissible n, the following equalities hold: (S+ `n )(y) = −`n+1 (y),

(S− `n )(y) = −`n−1 (y),

and

(S− `0 )(y) = 0.

Proof. By [6], formula 8.971.1, we have L0n (y) − L0n+1 (y) = Ln (y).

(8)

Taking into account that Ln (0) = 1, for all n, the integral form of the above formula is as follows: Z y Ln (y) − Ln+1 (y) = Ln (t)dt. 0

Calculate now

Z y

y

y

−e− 2 Ln (y) + e− 2

(S+ `n )(y) =

Ln (t)dt 0

y

e− 2 (−Ln (y) + Ln (y) − Ln+1 (y)) = −`n+1 (y).

=

Integrating by parts twice and using (8), we have Z ∞ Z ∞ Z e−t Ln (t)dt = e−y Ln (y) + e−t L0n−1 (t)dt − y

y

Z =

y −y

e−t Ln−1 (t)dt

y

∞

e−y Ln (y) −

∞

Z

e−t Ln−1 (t)dt ∞

Ln−1 (y) +

e−t Ln−1 (t)dt

−

e

=

e−y Ln (y) − e−y Ln−1 (y).

y

Thus (S− `n )(y) =

− y2

−e

Ln (y) + e

y 2

Z

∞

e−t Ln (t)dt

y

=

¢ y y ¡ −e− 2 Ln (y) + e 2 e−y Ln (y) − e−y Ln−1 (y) = −`n−1 (y).

Finally, y

y

Z

(S− `0 )(y) = −e− 2 + e 2

∞

e−t dt = 0.

y

¤ It is convenient to change the previously used basis {`n (y)}∞ n=0 of L2 (R+ ) to the new basis {èn (y)}∞ , where n=0 èn (y) = (−1)n `n (y),

n = 0, 1, 2, ... .


9

We note that the previously defined one-dimensional spaces Ln are generated by the new basis elements èn (y) as well, and that the statements of Theorem 2.1 remain valid without any change. Remark 3.3. As the previous theorem shows, the operator S+ is an isometric operator on L2 (R+ ) and is nothing but the unilateral forward shift with respect to the basis {èn (y)}∞ n=0 . Its adjoint operator S− is the unilateral backward shift with respect to the same basis, and its kernel coincides with the one-dimensional space y L0 generated by è0 (y) = e− 2 . The above, together with Theorem 3.1, permits us to give a simple functional ∗ . Each of them is unitary equivalent to the model for both operators SΠ and SΠ direct sum of two unilateral shifts, forward and backward, both taken with the infinite multiplicity. Let L⊕ n =

n M

Lk

k=0

be the direct sum of the first (n + 1) Lk -spaces. We denote by Pn and Pn⊕ the orthogonal projections of L2 (R+ ) onto Ln and L⊕ n , respectively. Corollary 3.4. For all admissible indices, we have P0 Pn

= =

I − S+ S− , n n , P0 S− S+

Pn⊕

=

n+1 n+1 I − S+ S− ,

k S+ |Ln

:

Ln −→ Ln+k ,

k S− |Ln

:

Ln −→ Ln−k .

The next result was obtained in [7] (see Theorem 2.4 and Corollary 2.6 ∗ therein) and shows that the action of both operators SΠ and SΠ is extremely transparent according to the decomposition L2 (Π) =

∞ M k=1

A2(k) (Π)

⊕

∞ M

Ae2(k) (Π).

k=1

In our approach it is just a straightforward corollary of Theorems 2.1, 3.1, and Corollary 3.4.

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N. Vasilevski

Theorem 3.5. For all admissible indices, we have (SΠ )k |A2

(Π)

(SΠ )k |Ae2

(Π)

∗ k (SΠ ) |Ae2

(Π)

∗ k (SΠ ) |A2

(Π)

(n)

(n)

(n)

(n)

:

A2(n) (Π) −→ A2(n+k) (Π),

:

Ae2(n) (Π) −→ Ae2(n−k) (Π),

:

Ae2(n) (Π) −→ Ae2(n+k) (Π),

:

A2(n) (Π) −→ A2(n−k) (Π),

ker(SΠ )n = Ae2n (Π),

(Im (SΠ )n )⊥ = A2n (Π), ∗ n ⊥ (Im (SΠ ) ) = Ae2n (Π).

∗ n ker(SΠ ) = A2n (Π),

Corollary 3.6. Each true-n-analytic function ψ admits the following representation, ψ = (SΠ )n−1 ϕ, where ϕ ∈ A2 (Π). Each true-n-anti-analytic function g admits the following representation, ∗ n−1 g = (SΠ ) f,

where f ∈ Ae2 (Π). eΠ,(n) the orthogonal projections of L2 (Π) onto the We denote by BΠ,(n) and B 2 2 e spaces A(n) (Π) and A(n) (Π), consisting of true-n-analytic and true-n-anti-analytic eΠ,n be the orthogonal projections of L2 (Π) functions respectively. Let BΠ,n and B 2 2 onto the spaces An (Π) and Aen (Π), consisting of n-analytic and n-anti-analytic functions respectively. We summarize now some important properties of the above projections in terms of singular operators. Theorem 3.7. For all admissible indices, we have BΠ eΠ B BΠ,n eΠ,n B ∗ n−1 BΠ,(n) = (SΠ )n−1 BΠ (SΠ ) ∗ n−1 eΠ,(n) = (SΠ ) eΠ (SΠ )n−1 B B BΠ,(n+1) eΠ,(n+1) B

=

∗ I − SΠ SΠ ,

= =

∗ SΠ , I − SΠ ∗ n I − (SΠ )n (SΠ ) ,

=

∗ n I − (SΠ ) (SΠ )n ,

=

∗ n−1 ∗ n (SΠ )n−1 (SΠ ) − (SΠ )n (SΠ ) ,

= =

∗ n−1 ∗ n (SΠ ) (SΠ )n−1 − (SΠ ) (SΠ )n , ∗ SΠ BΠ,(n) SΠ , ∗ e SΠ BΠ,(n) SΠ .

=

Proof. Follows directly from Theorems 2.1, 3.1, and Corollary 3.4.

¤


11

References [1] M. B. Balk. Polyanalytic functions and their generalizations, Complex analysis I. Encycl. Math. Sci, volume 85, pages 195–253. Springer Verlag, 1997. [2] M. B. Balk and M. F. Zuev. On polyanalytic functions. Russ. Math. Surveys, 25(5):201–223, 1970. [3] Harry Bateman and Arthur Erdélyi. Higher transcendental functions, Vol. 2. McGraw-Hill, 1954. [4] A. Dzhuraev. Multikernel functions of a domain, kernel operators, singular integral operators. Soviet Math. Dokl., 32(1):251–253, 1985. [5] A. Dzhuraev. Methods of Singular Integral Equations. Longman Scientific & Technical, 1992. [6] I. S. Gradshteyn and I. M. Ryzhik. Tables of Integrals, Series, and Products. Academic Press, New York, 1980. [7] Yu. I. Karlovich and Lu´ıs Pessoa. C ∗ -algebras of Bergman type operators with piecewise continuous coefficients. Submitted for publication. [8] Josue Ram´ırez and Ilya M. Spitkovsky. On the algebra generated by a poly-Bergman projection and a composition operator. Factorization, singular operators and related problems (Funchal, 2002), pages 273–289. Kluwer Acad. Publ., Dordrecht, 2003. [9] N. L. Vasilevski. On the structure of Bergman and poly–Bergman spaces. Integr. Equat. Oper. Th., 33:471–488, 1999. [10] N. L. Vasilevski. Toeplitz operators on the Bergman spaces: Inside-the-domain effects. Contemp. Math., 289:79–146, 2001. N. Vasilevski Departamento de Matem´ aticas, CINVESTAV del I.P.N., Apartado Postal 14-740, 07000 México, D.F., México e-mail: [email protected]