Two weights inequality for Hankel operators on weighted Bergman spaces induced by radial weights

TL;DR

This work characterizes two-weights inequalities for Hankel operators $H_f^{\omega}$ on radial-weighted Bergman spaces by providing necessary and sufficient conditions for boundedness and compactness of $H_f^{\omega}: A_v^p\to L_\eta^q$ under $1<p,q<\infty$ and $\omega\neq v\neq \eta\in \mathcal{R}$. The authors establish equivalent criteria in two regimes: (i) $1\le p\le q$ via a $G_{q,r}$-type control of the symbol and a holomorphic-decomposition $f=f_1+f_2$ with explicit decay/regularity properties, and (ii) $1<q<p$ via a weighted integrability condition $G_{q,r}(f)\in L_W^{pq/(p-q)}$ with $W=\eta^{p/(p-q)}v^{-q/(p-q)}$, supported by Khintchine-based randomization arguments. These results generalize earlier single-weight Hankel analyses (e.g., Hu 2019) to the two-weight setting with radial weights, hinging on Bergman kernel estimates, Carleson measure criteria, and partition-of-unity techniques on Bergman lattices. The findings advance the understanding of how symbol regularity and weight interactions govern operator boundedness and compactness in weighted analytic function spaces, with implications for operator theory on generalized Bergman spaces.

Abstract

The two weights inequality for Hankel operators $$\|H_f^ω(\cdot)\|_{L_η^q}\leq C \|\cdot\|_{A_v^p},$$ induced by some radial weights under the regular assumptions is considered, the boundedness and compactness of Hankel operators $H^ω_f$ is characterized for $1<p,q<\infty$ and $ω\neq v\neq η\in \mathcal{R}$.

Theorems & Definitions (18)

• Lemma A
• proof
• Lemma B
• Remark 1
• Lemma 2.1
• Lemma 2.2
• Remark 2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5