Mathematical Analysis/Harmonic Analysis

Reproducing kernels for harmonic Besov spaces on the ball^{}

Seçil Gergün^{a}, H. Turgay Kaptanoğlu^{b,∗}, A. Ersin Üreyen^{c}

aDepartment of Mathematics and Computer Science, Çankaya University, Ankara 06530, Turkey bDepartment of Mathematics, Bilkent University, Ankara 06800, Turkey cDepartment of Mathematics, Anadolu University, Eskişehir 26470, Turkey

Abstract

Besov spaces of harmonic functions on the unit ball of ℝ^{n} are defined by requiring sufficiently high-order derivatives of functions lie in harmonic Bergman spaces. We compute the reproducing kernels of those Besov spaces that are Hilbert spaces. The kernels turn out to be weighted infinite sums of zonal harmonics and natural radial fractional derivatives of the Poisson kernel.

Résumé

Noyaux reproduisants pour les espaces harmoniques de Besov sur la boule. Les espaces de Besov de fonctions harmoniques sur la boule unité de ℝ^{n} sont défini en exigeant que suffisamment des dérivés de haut ordre de fonctions appartiennent aux espaces de Bergman harmoniques. Nous calculons les noyaux reproduisants de ces espaces de Besov qui sont des espaces de Hilbert. Les noyaux se révèlent être, de façon tout naturel, des sommes infinies pondérées des harmoniques zonalles et des dérivés fractionnels radiaux du noyau de Poisson.

Key words: Spherical harmonic, zonal harmonic, Poisson kernel, reproducing kernel, radial fractional derivative, Besov space, Bergman space. 2000 MSC: 46E22, 31B05, 33C55, 26A33, 42B35.

1. Introduction and Preliminaries

Let 𝔹 and 𝕊 be the unit ball and the unit sphere in ℝ^{n} with respect to the usual inner product x · y = x_{1}y_{1} + ⋯ + x_{n}y_{n} and the norm |x| = \sqrt{x · x}. Always n ≥ 2, and we often write x = rξ, y = ρη with r = |x|, ρ = |y| and ξ, η ∈ 𝕊. We let ν and σ be the volume and surface measures on 𝔹 and 𝕊 normalized as ν(𝔹) = 1 and σ(𝕊) = 1. We always take q ∈ ℝ, and define on 𝔹 also the measures dν_{q}(x) = N_{q} (1 − |x|^{2})^{q} dν(x). These measures are finite only for q > −1 and then we choose N_{q} so as to have ν_{q}(𝔹) = 1. For q ≤ −1, we set N_{q} = 1. We denote the Lebesgue classes with respect to ν_{q} by L_{q}^{p}, and we always consider 1 ≤ p < ∞.

We let h(𝔹) denote the space of complex-valued harmonic functions on 𝔹, those annihilated by the usual Laplacian Δ, with the topology of uniform convergence on compact subsets of 𝔹. The Besov spaces under consideration in this Note form a two-parameter Sobolev-type family within h(𝔹). They extend Bergman spaces of harmonic functions. They have been studied early in [4]–[8] from a different perspective.

This research is supported by TÜBİTAK under Research Project Grant 108 T 329.

∗Corresponding author.

Email addresses: gergun@cankaya.edu.tr (Seçil Gergün), kaptan@fen.bilkent.edu.tr (H. Turgay Kaptanoğlu), aureyen@anadolu.edu.tr (A. Ersin Üreyen)

URL: http ://www.fen.bilkent.edu.tr/∼kaptan/ (H. Turgay Kaptanoğlu)

Preprint submitted to the Académie des sciences May 14, 2009

The Pochhammer symbol is defined by (a)_{b} := Γ(a + b)/Γ(a) when a and a + b are off the pole set of the gamma function Γ. Stirling formula gives

\frac{Γ(c + a)}{Γ(c + b)} ∼ c^{a−b}  and  \frac{(a)_{c}}{(b)_{c}} ∼ c^{a−b}  (Re c → ∞),  (1)

where x ∼ y means that |x/y| is bounded above and below by two positive constants.

The rest of the material in this section is classical and can be found in [1, Chapter 5] or [11, Chapter IV]. For m = 0, 1, 2, …, let H_{m} denote the space of all harmonic homogeneous polynomials of degree m. By homogenity, a u ∈ H_{m} is determined by its restriction to 𝕊, and we freely identify u with its restriction. The restrictions are called spherical harmonics. The space L^{2}(σ) is the orthogonal direct sum of all the H_{m}.

Lemma 1.1. If u ∈ h(𝔹), then u has a unique homogeneous expansion u = ∑_{m=0}^{∞} u_{m} with u_{m} ∈ H_{m}, the series converging absolutely on 𝔹, and uniformly on its compact subsets.

The spaces H_{m} are finite-dimensional; we set δ_{m} = dim H_{m}. Then the evaluation functionals at points η ∈ 𝕊 are bounded on H_{m}, and so H_{m} is a reproducing kernel Hilbert space. Its reproducing kernel Z_{m}(ξ, η) is called the zonal harmonic of degree m; thus Z_{m} is a positive definite function. Zonal harmonics can be extended to positive definite functions on \overline{𝔹} × \overline{𝔹} as Z_{m}(x, y) = r^{m} ρ^{m} Z_{m}(ξ, η) by homogeneity. Zonal harmonics are real-valued and symmetric in their variables, that is, Z_{m}(x, y) = Z_{m}(y, x) for x, y ∈ \overline{𝔹}. Consequently, Z_{m} is harmonic in either of its variables since it lies in H_{m}. There are explicit formulas for Z_{m}(x, y) in the two books mentioned above, but we do not need them in this Note.

Our main results are given in Section 3. Detailed proofs and further results will be presented elsewhere.

2. Harmonic Bergman Spaces and Kernels

For q > −1, the Bergman space b_{q}^{p} is that closed subspace of L_{q}^{p} whose members are also in h(𝔹). The Bergman spaces with p = 2 are reproducing kernel Hilbert spaces, and the reproducing kernel of b_{q}^{2} is

R_{q}(x, y) = ∑_{m=0}^{∞} \frac{(n/2 + 1 + q)_{m}}{(n/2)_{m}} Z_{m}(x, y) =: ∑_{m=0}^{∞} γ_{m}(q) Z_{m}(x, y)  (q > −1, x, y ∈ 𝔹),  (2)

which also defines γ_{m}(q); see [9, Proposition 3]; also [6, p. 25], [8], and [1, pp. 156–157] for q = 0; and [13, p. 357] for n = 2; and also [2, (3.1)] for integer q. The kernels R_{q} converge absolutely on 𝔹 × 𝔹, and uniformly if one variable lives in a compact subset of 𝔹. Therefore the R_{q} are symmetric in their variables and harmonic as a function of each.

The computation yielding R_{q} is valid only for q > −1, but R_{−1} also perfectly makes sense and is nothing but the homogeneous extension of the Poisson kernel P to 𝔹 × 𝔹 since γ_{m}(−1) = 1 for all m. In fact, the coefficients γ_{m}(q) make sense down to q > −(n/2 + 1), and for all such q, they satisfy

γ_{m}(q) ∼ m^{1+q}  (m → ∞)  (3)

by (1). With smaller γ_{m}(q), the infinite sums in (2) for q ≤ −1 have at least the same convergence properties on 𝔹 × 𝔹 as those for q > −1. Since γ_{m}(q) > 0 for all m and q > −(n/2 + 1), and the Z_{m} are positive definite kernels, by convergence we conclude that R_{q} given as in (2) is a positive definite function, and thus is a reproducing kernel and generates a reproducing kernel Hilbert space on 𝔹 for all q > −(n/2 + 1). All the references cited in the previous paragraph restrict themselves to q > −1 or so. The point of view of getting from kernels to spaces apparently has never been utilized before.

3. Harmonic Besov Kernels and Spaces

Our purpose now is to extend the kernels R_{q} even further to all q ∈ ℝ. The main idea is to replace the coefficients γ_{m}(q) of Z_{m} in R_{q} by new γ_{m}(q) that preserve the growth rate of (3) for q ≤ −(n/2 + 1) as well.

Definition 3.1. For m = 0, 1, 2, …, we set

γ_{m}(q) = \begin{cases}\frac{(n/2 + 1 + q)_{m}}{(n/2)_{m}} , & if q > −(n/2 + 1); \\ \frac{(m!)^{2}}{(1 − n/2 − q)_{m} (n/2)_{m}} , & if q ≤ −(n/2 + 1); \end{cases}

and define

R_{q}(x, y) = ∑_{m=0}^{∞} γ_{m}(q) Z_{m}(x, y).  (4)

It seems the kernels (4) for q ≤ −(n/2 + 1) are completely new here. Note that γ_{0}(q) = 1 for all q and (3) is satisfied for all real q by (1) as promised. Moreover, γ_{m}(q) ≠ 0 for all m = 0, 1, 2, … and all q ∈ ℝ.

Proposition 3.2. The series (4) converges absolutely for any x, y ∈ 𝔹. If one of x or y lies in a compact subset of 𝔹 and the other in \overline{𝔹}, then the series converges uniformly.

Definition 3.3. Let u = ∑_{m=0}^{∞} u_{m} ∈ h(𝔹) be given as in Lemma 1.1. We define operators D_{s}^{t} by

D_{s}^{t}u := ∑_{m=0}^{∞} d_{m}(s, t) u_{m} := ∑_{m=0}^{∞} \frac{γ_{m}(s + t)}{γ_{m}(s)} u_{m}.

For any s, D_{s}^{0} = I, the identity. If λ is a multi-index, then D_{s}^{t}x^{λ} = d_{|λ|}(s, t) x^{λ}. So in every case D_{s}^{t}(H_{m}) = H_{m}. By Definition 3.1 and (1), we have d_{m}(s, t) ∼ m^{t} as m → ∞ for all s ∈ ℝ. Particularly, D_{−n/2}^{1} = R + I, where R is the radial derivative given by Ru(x) = ∇u(x) · x = ∑_{m=0}^{∞} m^{1} u_{m}(x) in which ∇ is the usual gradient. Summing up, each D_{s}^{t} is a radial differential operator of fractional order t.

Moreover, d_{m}(s, t) ≠ 0 for all choices of m, s, t. Then every D_{s}^{t} is invertible with two-sided inverse (D_{s}^{t})^{−1} = D_{s+t}^{−t}, which follows from the additive property D_{s+t}^{u}D_{s}^{t} = D_{s}^{t+u}. The operators D_{s}^{t} are constructed so that D_{s}^{t}R_{s}(x, y) = R_{s+t}(x, y) in all cases, where differentiation is performed only on one of the variables x, y; and by symmetry it does not matter which. In particular, R_{q}(x, y) = D_{−1}^{1+q}P (x, y) extending [2, (3.1)], [1, 8.12], and a formula on [9, p. 29].

Lemma 3.4. Every D_{s}^{t} maps h(𝔹) into itself continuously. Thus D_{s}^{t}u is harmonic on 𝔹 if u is.

Definition 3.5. Consider the linear transformation I_{s}^{t} defined for u ∈ h(𝔹) by I_{s}^{t}u(x) = (1 − |x|^{2})^{t} D_{s}^{t}u(x). For q ∈ ℝ and 1 ≤ p < ∞, we define the harmonic Besov space b_{q}^{p} to consist of all u ∈ h(𝔹) for which I_{s}^{t}u belongs to L_{q}^{p} for some s, t satisfying

q + pt > −1.  (5)

Condition (5) assures that all b_{q}^{p} contain the polynomials and therefore are nontrivial. Thus H_{m} ⊂ b_{q}^{p} for all possible values of the parameters. For any s, t satisfying (5), by the invertibility of D_{s}^{t} and that 1 − |x|^{2} ≠ 0 for x ∈ 𝔹, the map I_{s}^{t} imbeds b_{q}^{p} into L_{q}^{p}. Then ∥u∥_{b_{q}^{p}} := ∥I_{s}^{t}f∥_{L_{q}^{p}} defines a norm on b_{q}^{p} for each such s, t, and only 0 ∈ h(𝔹) has norm 0. Similarly, each pair s, t satisfying (5) with p = 2 gives rise to an inner product on b_{q}^{2} by [u, v]_{b_{q}^{2}} = [I_{s}^{t}u, I_{s}^{t}v]_{L_{q}^{2}}. Explicitly,

∥u∥_{b_{q}^{p}}^{p} = N_{q} ∫_{𝔹} |D_{s}^{t}u(x)|^{p} (1 − |x|^{2})^{q+pt} dν(x), [u, v]_{b_{q}^{2}} = N_{q} ∫_{𝔹} D_{s}^{t}u(x) \overline{D_{s}^{t}v(x)} (1 − |x|^{2})^{q+2t} dν(x).

Proposition 3.6. Different values of s, t satisfying (5) give equivalent norms ∥ · ∥_{b_{q}^{p}} on b_{q}^{p}.

Definition 3.5 assigns the space b_{q}^{p} to the point (p, q) in the half plane {Re p ≥ 1}. When q > −1, we can take t = 0, and thus the spaces b_{q}^{p} are the well-known harmonic weighted Bergman spaces. Our main interest lies in the region q ≤ −1, but our results cover and generalize what is known for q > −1 as well.

Harmonic Besov spaces are studied in [4]–[8] in the generality of Definition 3.5, but recently only smaller subfamilies are considered such as with q = −n or t = 1 or both; see [3, 10, 12, 14].

Theorem 3.7. Each b_{q}^{2} for q ∈ ℝ is a reproducing kernel Hilbert space, and its reproducing kernel is R_{q}.

For another description of b_{q}^{2}, let’s start with the homogeneous expansion of u ∈ h(𝔹) given in Lemma 1.1. If {Y_{m1}, …, Y_{mδ_{m}}} is an orthonormal basis for H_{m} ⊂ L^{2}(σ), then each u_{m} restricted to 𝕊 has itself an expansion in terms of the {Y_{mk}}, and thus

u(x) = ∑_{m=0}^{∞} r^{m} u_{m}(ξ) = ∑_{m=0}^{∞} r^{m} ∑_{k=1}^{δ_{m}} c_{mk} Y_{mk}(ξ) = ∑_{m=0}^{∞} ∑_{k=1}^{δ_{m}} c_{mk} Y_{mk}(x)  (x ∈ 𝔹),  (6)

c_{mk} = ∫_{𝕊} u_{m}(ξ) \overline{Y_{mk}(ξ)} dσ(ξ) = \frac{1}{r^{m}} ∫_{𝕊} u_{m}(rξ) \overline{Y_{mk}(ξ)} dσ(ξ) = \frac{1}{r^{m}} ∫_{𝕊} u(rξ) \overline{Y_{mk}(ξ)} dσ(ξ) (0 < r < 1)

by orthogonality. We also see that this computation of c_{mk} is independent of r ∈ (0, 1).

Theorem 3.8. The Hilbert space b_{q}^{2} coincides with the space β_{q} of functions u ∈ h(𝔹) with expansions of the form (6) for which

∑_{m=0}^{∞} ∑_{k=1}^{δ_{m}} \frac{|c_{mk}|^{2}}{γ_{m}(q)} < ∞  (7)

equipped with the inner product ⟨u, v⟩_{q} = ∑_{m=0}^{∞} ∑_{k=1}^{δ_{m}} \frac{1}{γ_{m}(q)} c_{mk} \overline{c′_{mk}} and the associated norm ∥|u|∥_{q} = ⟨u, u⟩^{1/2},

where primes denote the coefficients of v ∈ h(𝔹).

Corollary 3.9. The norms ∥| · |∥_{q} and ∥ · ∥_{b_{q}^{2}} are equivalent on b_{q}^{2}.

Example 3.10. Harmonic Besov spaces b_{q}^{2} with different q are different. Let q_{1} < q_{2}. By Definition 3.5, it is clear that b_{q_{1}}^{2} ⊂ b_{q_{2}}^{p}. Next, define u(x) = ∑_{m=1}^{∞} m^{(q_{1}+q_{2})/4} Y_{m1}(x). Then u ∈ b_{q_{2}}^{2} \ b_{q_{1}}^{2}, because by (7),

∥|u|∥_{q_{1}} ∼ ∑_{m=1}^{∞} \frac{1}{m^{1−(q_{2}−q_{1})/2}} = ∞  while  ∥|u|∥_{q_{2}} ∼ ∑_{m=1}^{∞} \frac{1}{m^{1+(q_{2}−q_{1})/2}} < ∞.

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