Harmonic Besov spaces on the ball

Abstract

We initiate a detailed study of two-parameter Besov spaces on the unit ball of [Formula: see text] consisting of harmonic functions whose sufficiently high-order radial derivatives lie in harmonic Bergman spaces. We compute the reproducing kernels of those Besov spaces that are Hilbert spaces. The kernels are weighted infinite sums of zonal harmonics and natural radial fractional derivatives of the Poisson kernel. Estimates of the growth of kernels lead to characterization of integral transformations on Lebesgue classes. The transformations allow us to conclude that the order of the radial derivative is not a characteristic of a Besov space as long as it is above a certain threshold. Using kernels, we define generalized Bergman projections and characterize those that are bounded from Lebesgue classes onto Besov spaces. The projections provide integral representations for the functions in these spaces and also lead to characterizations of the functions in the spaces using partial derivatives. Several other applications follow from the integral representations such as atomic decomposition, growth at the boundary and of Fourier coefficients, inclusions among them, duality and interpolation relations, and a solution to the Gleason problem.

Precise inclusion relations among Bergman–Besov and Bloch–Lipschitz spaces and on the unit ball of

“…Note that , for any k , and for any by . So is a continuous operator on and is of order t ; for a proof of a similar continuity result, see [, Theorem 3.2]. In particular, for any multi‐index γ, and hence .…”

“…(ii) Assume now and . The proof is a variant of those of [, Proposition 13.2] and part (i). For any , the finiteness of the measure gives us the other fundamental inclusion .…”

“…Proof of Theorem (i) Assume and . We follow the method of [, Proposition 13.3] to obtain the desired inclusion. First let satisfy with and p ; then clearly satisfy with .…”

Harmonic Besov spaces with small exponents

“…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).…”