Reproducing kernels for harmonic Besov spaces on the ball

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.

Harmonic Besov spaces with small exponents

“…for all such k by (9). The reproducing kernels R α (x, y) can be extended to all α ∈ R (see [9,10]), where the crucial point is not the precise form of the kernel but preserving the property (11).…”

“…Checking the two cases above by (9), the property (11) holds for all α ∈ R. R α (x, y) given as in Definition 2.2 is a reproducing kernel and generates the reproducing kernel Hilbert space b 2 α on B for all α ∈ R (see [10]). For every α ∈ R we have γ 0 (α) = 1 and therefore with Lemma 2.1 (b),…”

“…We study in a detailed and systematic way the properties of this family. For the case 1 ≤ p < ∞ and α ∈ R the spaces b p α are studied in [9,10]. The holomorphic counterpart of this family of spaces for the full range of parameters have been studied in [17].…”

“…The aim of this work is to extend the definition of b p α when 0 < p < 1 to the case in which α is any real number and develop a theory for the extended family of spaces. This family of spaces are already extended to all α ∈ R for the case 1 ≤ p < ∞ in [9,10] where also the reproducing kernels R α (x, y) are extended to the whole range α ∈ R. We will give a review of these in Section 2. 2 To extend the definition of b p α to the range α ≤ −1, we need to consider growth rates of derivatives of u ∈ h(B).…”

“…For any u ∈ h(B) there exist homogeneous harmonic polynomials u k of degree k such that u = ∞ k=0 u k , the series converging absolutely and uniformly on compact subsets of B which is called the homogeneous expansion of u (see [2]). The radial derivative Ru of u ∈ h(B) is defined as (1) Ru(x) = x · ∇u(x) = In addition to partial and radial derivatives we will extensively use certain radial differential operators D t s , (s, t ∈ R) introduced in [9] and [10]. These operators are defined in terms of reproducing kernels of harmonic Besov spaces and are specific to these spaces, but still mapping h(B) onto itself.…”