Da Lio-Rivière-Wettstein-type inequality for weighted Bergman spaces

TL;DR

This work extends the Da Lio-Rivière-Wettstein-type framework to radial weighted Bergman spaces, establishing a sharp boundary-value characterization on the unit disk: $$(B^2(\mathbb{D}, \mu) + h^1(\mathbb{D})) \cap \mathrm{Hol}(\mathbb{D}) = A^2(\mathbb{D}, \mu)$$ holds precisely when $\mu$ is a $$(1,2)$$-Carleson measure; an analogous formulation holds for the upper half-plane via Zen-type spaces. The authors prove a weighted Bourgain-Brezis inequality using a Fourier-m multiplier $\mathcal{A}_\mu$ tied to the weight’s moment sequence, and exploit Carleson measure characterizations to transfer estimates between harmonic and holomorphic spaces. They also address holomorphic stability and complemented subspaces, deriving conditions under which the disk and upper-half-plane problems are equivalent or complemented, and provide a comprehensive treatment of the horizontal-invariant upper-half-plane case with detailed operator decompositions. Overall, the paper clarifies when weighted Bergman spaces admit clean boundary characterizations and stable decompositions, enriching the theory of weighted projections and their interaction with Carleson-type conditions in both planar domains.

Abstract

In this paper, inspired by the work of Da Lio-Rivière-Wettstein, we investigate the boundary-value characterizations of weighted Bergman spaces and establish a weighted Da Lio-Rivière-Wettstein inequality. In addition, we obtain analogous results on the upper plane which does not seem to be a direct consequence of the ones on the unit disk.

Theorems & Definitions (29)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['thm-disk-stable']}
• Proposition 3.2