Embedding theorems for Bergman spaces via harmonic analysis

Abstract

Let $A^p_\omega$ denote the Bergman space in the unit disc induced by a radial weight~$\omega$ with the doubling property $\int_{r}^1\omega(s)\,ds\le C\int_{\frac{1+r}{2}}^1\omega(s)\,ds$. The positive Borel measures such that the differentiation operator of order $n\in\mathbb{N}\cup\{0\}$ is bounded from $A^p_\omega$ into $L^q(\mu)$ are characterized in terms of geometric conditions when $0<p,q<\infty$. En route to the proof a theory of tent spaces for weighted Bergman spaces is built.Comment: appears in Mathematische Annalen 201

On the boundedness of Bergman projection

“…Next, we present an equivalent norm for weighted Bergman spaces which has been very recently used to describe the q-Carleson mesures for A p ω when ω ∈ D [31].…”

Besov Spaces Induced by Doubling Weights

“…In this section, we recall some basic properties of weights in p D and q D. These properties are needed in next sections. Another reason for these results is to help the reader to understand the nature of weights in D. We begin with a result which is essentially [19,Lemma 3]; see also [18…”

“…As a concrete example, we mention that ν 1 pzq " p1´|z|q α and ν 2 pzq " p1| z|q α´l og e 1´|z|¯β for any β P R belong to D X p D p if and only if´1 ă α ă p´1. Additional information about weights can be found in [18,19,20]. Some basic properties are recalled also in Section 2.…”

Derivatives of inner functions in Bergman spaces induced by doubling weights

“…It is known that the existence of β such that the right-hand inequality is satisfied is equivalent to ω ∈ D by [29,Lemma 1], and therefore D = ∪ p>0 D p . It is easy to see that the left-hand inequality is equivalent to the existence of K = K(ω) > 1 and C = C(ω) > 1 such that the doubling property ω(r) ≥ C ω 1 − 1−r K is satisfied for all 0 ≤ r < 1.…”