The Polyharmonic Bergman Space for the Union of Rotated Unit Balls

In the paper we consider the polyharmonic Bergman space for the union of the rotated unit Euclidean balls. Using so called zonal polyharmonics we derive the formulas for the kernel of this space. Moreover, we study the weighted polyharmonic Bergman space. By the same argument we get the Bergman kernel for this space.


Introduction
The polyharmonic Bergman spaces have recently been extensively studied (see [7,8,11] or [12]). They are mainly considered on the unit ball or on its complement. However, we regard the space of polyharmonic and square integrable functions on the set B p = p−1 k=0 e kπi p B (in fact, we may assume the polyharmonicity only on B, because every polyharmonic function can be extended analytically from B onto every rotated Euclidean ball). More precisely, we consider the space of the polyharmonic functions on B p such that  B Hubert Grzebuła h.grzebula@student.uksw.edu.pl 1 Such space we denote by b 2 p ( B p ). The motivation to study the polyharmonic functions on such set is given in the paper [3] (see also [5] and [6]).
Using the mean value property for polyharmonic functions (see Lemma 5 in [7]) and some theorems, we conclude that b 2 p ( B p ) is a Hilbert space with the inner product u(e kπi p y)v(e kπi p y) dy.
Further, by theorem of Riesz, there exists a function R p (x, ·) ∈ b 2 p ( B p ) such that u(x) = u, R p (x, ·) b 2 p for every u ∈ b 2 p ( B p ). The function R p (x, ·) is a reproducing kernel for the Bergman space and it is called the Bergman kernel for B p . Using some properties of spherical polyharmonics and zonal polyharmonics we get the formula for the Bergman kernel which is similar to the harmonic Bergman kernel: where n = π n/2 / (n/2 + 1) is the volume of the unit ball B in R n . By the formula for polyharmonic Poisson kernel (see Theorem 4 in [4]) we can express the Bergman kernel in the term of polyharmonic Poisson kernel P p (x, y) R p (x, y) = 1 n n n P p (x, y) + d dt P p (t x, t y) t=1 and from this we obtain the explicit formula for Bergman kernel R p (x, y) = (n − 4 p)|x| 2 p+2 |y| 2 p+2 + (8 px y − n − 4 p)|x| 2 p |y| 2 p + n(1 − |x| 2 |y| 2 ) n n (1 − 2x y + |x| 2 |y| 2 ) n/2+1 .
Moreover, we can express R p (x, y) in the terms of the harmonic Bergman kernel R(x, y) and the harmonic Poisson kernel P(x, y): Next we consider the weighted polyharmonic Bergman space. Here we study polyharmonic functions on B p , which satisfy the following condition where n + α > 0, β > −1. We denote this space by b 2 p,α,β ( B p ). By the similar arguments we prove that b 2 p,α,β ( B p ) is a Hilbert space and there exists the reproducing kernel R p,α,β (x, ·) ∈ b 2 p,α,β ( B p ) such that The function R p,α,β is called a polyharmonic weighted Bergman kernel. We get the formula for this kernel The paper is organised as follows. In the next section we give some basic notations and one lemma about extension of polyharmonic functions from the real ball onto its rotation (Lemma 1).
In the third section we recall some informations about the spherical polyharmonics, zonal polyharmonics, polyharmonic Poisson kernel and their properties (Lemmas 2-6). By these lemmas we get another properties for polyharmonic functions (Propositions 1 and 2).
In the next section we introduce the polyharmonic Bergman space. Using Lemma 7 we get some properties for this space (Propositions 3 and 4).
In the fifth section we introduce the polyharmonic Bergman kernel for the set B p . We give basic properties for this function (Proposition 5) and some another properties for the polyharmonic functions (Propositions 6-8). Using these properties and those ones given in Sect. 3 we get the formula for the polyharmonic Bergman kernel (Theorem 1). Moreover we get the explicit form for this function (Theorem 2) and we express it in the terms of harmonic Bergman kernel and harmonic Poisson kernel (Theorem 3).
In the sixth section we consider the polyharmonic weighted Bergman space. Similarly as for unweighted one we show that this space is a Hilbert space and there exists the reproducing kernel called the weighted Bergman kernel (Corollary 1). By similar arguments (Lemma 9, Propositions 9 and 10), we get the formulas for this kernel (Theorems 4-6).

Preliminaries
In this section we give some basic notations and definitions.
We define the real norm with |z j | 2 C = z j z j . We will also use the complex extension of the real norm for complex vectors:

By
x y = x 1 y 1 + x 2 y 2 + · · · + x n y n we denote the usual inner product for the complex (real) vectors x, y.
By a square root in the above formula we mean the principal square root, where a branch cut is taken along the non-positive real axis. Obviously the function | · | is not a norm in C n , because it is complex valued and hence the function |z − w| is not a metric on C n .
We will consider mainly complex vectors of the form z = e iϕ x, that is vectors x ∈ R n rotated in C n by the angle ϕ.
For the set G ⊆ R n and the angle ϕ ∈ R we will consider the rotated set defined by We will consider mainly the following unions of rotated sets in C n : where B and S are respectively the unit ball and sphere in R n with a centre at the origin.
Let G be an open set in R n . We denote by A(G) the space of analytic functions on G. Similarly we say that f ∈ A(e iϕ G) if and only if f ϕ (x) := f (e iϕ x) ∈ A(G). We call the space of harmonic functions on G, where x denotes the Laplacian in R n . Analogously we define the family A (e iϕ G) of harmonic functions on e iϕ G. Observe that f ∈ A (e iϕ G) if and only if f ϕ ∈ A (G). Similarly, replacing the Laplace operator x by its pth iteration p x in the above definitions, we introduce the spaces of polyharmonic functions of degree p, that is A p (G) and A p (e iϕ G).

Lemma 1 [3, Lemma 1]
Let ϕ ∈ R and u ∈ A p (B). Then the function u has a holomorphic extension to the set {z ∈ C n : z = e iψ x, ψ ∈ R, x ∈ B}, whose restriction u ϕ to e iϕ B is polyharmonic of order p, i.e. u ϕ ∈ A p (e iϕ B).

Zonal Polyharmonics and Polyharmonic Poisson Kernel
In this section we recall the spherical and zonal polyharmonics and the polyharmonic Poisson kernel (see [4]). Let m, p ∈ N. We denote by H p m (C n ) the space of polynomials on C n , which are homogeneous of degree m and are polyharmonic of order p. By homogeneous polynomial of degree m we mean the polynomial q such that q(az) = a m q(z) for every a ∈ C and for every z ∈ C n .
Let's observe that if m < 2 p, then H The spherical polyharmonics of order 1 are called spherical harmonics and their . We shall recall some properties of spherical polyharmonics. To do this let us consider the Hilbert space L 2 ( S p ) of square-integrable functions on S p with the inner product defined by where dσ is a normalized surface-area measure on the unit sphere S.

Lemma 2 [4, Propositions 4] The space H
It means that where the sum is converging in the norm of L 2 ( S p ).

By Lemma 3 we may consider H
Let η ∈ S p be a fixed point. Let us consider the linear functional η : Definition 2 [4, Definition 2] The function Z p m (·, η) satisfying (2) is called a zonal polyharmonic of degree m and of order p with a pole η.
Zonal polyharmonics of order p = 1 are called zonal harmonics. Throughout this paper we will denote them by Z m (·, η) instead of Z 1 m (·, η) for η ∈ S. Let's observe that we can extend the definition of zonal harmonics from S × S on S p × S p as follows: for any ζ, η ∈ S and j, l = 0, 1, . . . , p − 1. Moreover, the zonal harmonics are extended on B × B (see 8.7 in [1]) and hence, by Lemma 1, they are extended on Let's give some properties of the zonal polyharmonics.
We may extend the zonal polyharmonics from S p × S p on B p × B p in the same way as for the zonal harmonics, therefore by Lemma 4 we have In particular so by (2) and homogeneity we may write The zonal polyharmonics give us the construction of the polyharmonic Poisson kernel.

Lemma 5 [4, Theorem 4] The Poisson kernel has the expansion
The series converges absolutely and uniformly on K × S p , where K is a compact subset of B p .
Since the Poisson kernel for B p is polyharmonic so continuous, there exists constant M > 0 such that for every x ∈ K , where K is a compact subset of B p . Therefore Since u n ⇒ u on every compact subset of B p , we can take the limit under the integral sign. Hence we conclude that so u is polyharmonic on B p . At the end let us observe that by (3) and (6) we may extend the polyharmonic Poisson kernel: Let's note that by the above considerations and the Lemma 1 we can also extend the harmonic Poisson kernel P(x, y) onto B p × B p .

The Bergman Space
Let us consider the polyharmonic functions of order p on B p such that The above condition makes sense because every polyharmonic function can be extended onto any rotated ball by Lemma 1. The set of polyharmonic functions that satisfy the above condition is called the polyharmonic Bergman space, abbreviated the Bergman space, and we denote it by b 2 When p = 1, we have the classical harmonic Bergman space b 2 (B) (see Chapter 8 in [1]). As in the harmonic case, we want to show that b 2 p ( B p ) is also a Hilbert space. To do this we will use the following mean value property for polyharmonic functions: Proof The proof follows from Proposition 3, it is analogous as in the harmonic case (see [1,Corollary 8.3]).

The Bergman Kernel
Let x ∈ B p be a fixed point. Let's consider the linear functional is a Hilbert space with the inner product (9), by Riesz theorem, there exists a unique function The function R(x, ·) := R 1 (x, ·) is the harmonic Bergman kernel and this function is given by (see Theorem 8.9 in [1]): Using Lemma 1, we extend R(x, y) on B p × B p (we can also use the fact that the functions Z m (x, y) are extended on B p × B p , see Sect. 3).

Proposition 5
The Bergman kernel has the following properties: Proof The proof is almost the same as in the harmonic case (see [1,Proposition 8.4]).

Theorem 1 The polyharmonic Bergman kernel is given by
where the series converges absolutely and uniformly on K × B p for every compact subset K ⊂ B p .
Proof By (10) and Propositions 7 and 8, we need only to show the convergence of the series. By Lemma 8 there is what completes the proof.

Theorem 2
The polyharmonic Bergman kernel is given by:
Adding the last equalities we get the desired formula.
In the next theorem we express the polyharmonic Bergman kernel in the terms of the harmonic Bergman kernel and harmonic Poisson kernel.

Theorem 3 The polyharmonic Bergman kernel is given by
Proof By Theorem 1 and (3) we have Using (8) and Theorem 1 with p = 1 we obtain 4k|x| 2k |y| 2k P(x, y).

The Weighted Bergman Kernel
Let α + n > 0 and β > −1. Let us consider the set of polyharmonic functions of order p on the ball B such that This space is called a polyharmonic weighted Bergman space with weights α, β and we denote it by b 2 p,α,β ( B p ). Hence where L 2 ( B p , |y| α (1 − |y| 2 ) β dy) is the space of measurable functions on B p which satisfy (12). By mean value property for polyharmonic functions (Lemma 7) we conclude that Lemma 9 [11,Introduction] Let n + α > 0, β > −1, then for every compact subset K ⊂ B and x ∈ K , there exists a constant C = C(K , n, p) such that Proposition 9 Let n + α > 0, β > −1, then for every compact subset K ⊂ B and x ∈ K , there exists a constant C = C(K , n, p) such that Proof The proof follows from Lemma 9 and it is similar to the proof of Proposition 3.
By the last corollary we conclude that b 2 p,α,β ( B p ) is a Hilbert space with the inner product (13). Again, let x ∈ B p be a fixed point and let the linear functional : b 2 p,α,β ( B p ) → C be such that x (u) = u(x), then by Corollary 1 and Riesz Theorem there exists the function R p,α,β ∈ b 2 p,α,β ( B p ) such that for every u ∈ b 2 p,α,β ( B p ) we have Hence By homogeneity (see Remark 1) we get = n + 2m + α + 2β + 2 n + 2m + α → 1 as m → ∞ and this completes the proof.
We may also give the counterpart of Theorem 2 using the fractional derivatives in the Riemann-Liouville sense (see for example [9]). Let's recall the definitions.
Let l > 0, then the primitive of u ∈ L 1 (0, 1) is as follows The derivative of order l is as follows where j is an integer number such that j − 1 ≤ l ≤ j. As in harmonic case (see [9]), using the identity D l+1 t k = (k + 1) (k − l) t k−l−1 and again the formula (8), we obtain the following theorem for polyharmonic case: |x| 2k |y| 2k Z m (x, y).
Using Theorem 4 we obtain the desired formula.