The essential norm of Toeplitz operators between Bergman spaces induced by doubling weights
Peiying Huang, Guangfu Cao
TL;DR
This paper addresses the essential-norm problem for Toeplitz operators between Bergman spaces induced by doubling weights on the unit ball, specifically $\mathcal{T}_\mu:A_\omega^p\to A_\omega^q$ with $1<p\le q<\infty$ and $\omega\in\mathcal{D}$. It derives sharp essential-norm estimates $\|\mathcal{T}_\mu\|_e$ via Carleson-block techniques and reproducing-kernel analysis, showing $\|\mathcal{T}_\mu\|_e \simeq \limsup_{|z|\to1} \frac{\mu(S_z)}{\omega(S_z)^{1+\frac{1}{p}-\frac{1}{q}}}$. The work further proves a boundedness-to-compactness dichotomy in the regime $1<q<p<\infty$, where boundedness implies compactness and hence a vanishing essential norm. By extending classical unweighted results and handling radial and non-radial doubling weights in higher dimensions, the paper provides a unified framework linking Carleson-measure conditions with operator-norm criteria for Toeplitz operators on weighted Bergman spaces.
Abstract
This paper investigates the essential norm of Toeplitz operators $\mathcal{T}_μ$ acting from the Bergman space $A_ω^p$ to $A_ω^q$ ($1 < p \leq q < \infty$) on the unit ball, where $μ$ is a positive Borel measure and $ω\in \mathcal{D}$ (a class of doubling weights). Leveraging the geometric properties of Carleson blocks and the structure of radial doubling weights, we establish sharp estimates for the essential norm in terms of the asymptotic behavior of $μ$ near the boundary. As a consequence, we resolve the boundedness-to-compactness transition for these operators when $1 < q < p<\infty$, showing that the essential norm vanishes exactly. These results generalize classical theorems for the unweighted Bergman space ($ω\equiv 1$) and provide a unified framework for studying Toeplitz operators under both radial and non-radial doubling weights in higher-dimensional settings.