Title: A local weighted Axler-Zheng theorem in $\mathbb{C}^n$,
Abstract: The well-known Axler-Zheng theorem characterizes compactness of finite sums
of finite products of Toeplitz operators on the unit disk in terms of the
Berezin transform of these operators. Subsequently this theorem was generalized
to other domains and appeared in different forms, including domains in
$\mathbb{C}^n$ on which the $\overline{\partial}$-Neumann operator $N$ is
compact. In this work we remove the assumption on $N$, and we study weighted
Bergman spaces on smooth bounded pseudoconvex domains. We prove a local version
of the Axler-Zheng theorem characterizing compactness of Toeplitz operators in
the algebra generated by symbols continuous up to the boundary in terms of the
behavior of the Berezin transform at strongly pseudoconvex points. We employ a
Forelli-Rudin type inflation method to handle the weights.