Harmonic Besov spaces with small exponents

Abstract

We study harmonic Besov spaces b p α on the unit ball of R n , where 0 < p < 1 and α ∈ R. We provide characterizations in terms of partial and radial derivatives and certain radial differential operators that are more compatible with reproducing kernels of harmonic Bergman-Besov spaces. We show that the dual of harmonic Besov space b p α is weighted Bloch space b ∞ β under certain volume integral pairing for 0 < p < 1 and α, β ∈ R. Our other results are about growth at the boundary and atomic decomposition.

A class of integral operators from Lebesgue spaces into harmonic Bergman-Besov or weighted Bloch spaces

“…Definition 2.2. For 0 < p < ∞ and α ∈ R, we define the harmonic Bergman-Besov space b p α to consist of all f ∈ h(B) for which I t s f belongs to L p α for some s, t satisfying (see [7] when 1 ≤ p < ∞, and [3]…”

Inclusion relations between harmonic Bergman-Besov and weighted Bloch spaces on the unit ball

“…Given 0 < p < ∞ and α ∈ R, pick s, t ∈ R such that α + pt > −1. The harmonic Bergman-Besov space b p α consists of all f ∈ h(B) such that (see [12] for 1 p < ∞, and [8]…”

“…For a proof of part (a) of the above lemma see [12], Corollary 9.2 for 1 p < ∞ and [8] for 0 < p < 1. For part (b) see [9], Proposition 4.6.…”

“…In the definition of b p α , instead of partial derivatives one can use radial derivatives or more effectively certain radial differential operators D t s defined in terms of reproducing kernels of harmonic Bergman spaces. These matters are studied in detail in [12] for 1 p < ∞ and in [8] for 0 < p < 1 and will be reviewed in Section 2.…”

“…. For 0 < p < ∞ and α ∈ R, we define the harmonic Bergman-Besov space b p α to consist of all f ∈ h(B) for which I t s f belongs to L p α for some s, t satisfying (see [12] when 1 ≤ p < ∞, and [5] when 0 < p < 1)…”