Weighted theory of Toeplitz operators on the Bergman space

Abstract

We study the weighted compactness and boundedness properties of Toeplitz operators on the Bergman space with respect to Békollè-Bonami type weights. Let $T_u$ denote the Toeplitz operator on the (unweighted) Bergman space of the unit ball in ${\mathbb{C}}^n$ with symbol $u\in L^{\infty }$. We characterize the compact Toeplitz operators on the weighted Bergman space ${\mathcal{A}}_{\sigma }^p$ for all $\sigma$ in a subclass of the Békollè-Bonami class $B_p$ that includes radial weights and powers of the Jacobian of biholomorphic mappings. Concerning boundedness, we show that $T_u$ extends boundedly on $L_{\sigma }^p$ for $p\in (1,\infty )$ and weights $\sigma$ in a $u$-adapted class of weights containing $B_p$, and we establish analogous weighted endpoint weak-type $(1,1)$ bounds for weights beyond $B_1$.

arXiv comment29 pages