The essential norm of Toeplitz operators between Bergman spaces induced by doubling weights

Abstract

This paper investigates the essential norm of Toeplitz operators $\mathcal{T}_\mu$ acting from the Bergman space $A_\omega^p$ to $A_\omega^q$ ($1 < p \leq q < \infty$) on the unit ball, where $\mu$ is a positive Borel measure and $\omega \in \mathcal{D}$ (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 $\mu$ 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 ($\omega \equiv 1$) and provide a unified framework for studying Toeplitz operators under both radial and non-radial doubling weights in higher-dimensional settings.