On Toeplitz operators on Bergman spaces of the unit polydisk

We study Toeplitz operators on the Bergman space $A^2_{\vartheta}$ of the unit polydisk $\mathbb{D}^n$, where $\vartheta$ is a product of $n$ rotation-invariant regular Borel probability measures. We show that if $f$ is a bounded Borel function on $\mathbb{D}^n$ such that $F(w)=\lim\limits_{\substack{z\rightarrow w z\in\mathbb{D}^n}}f(z)$ exists for all $w\in\partial\mathbb{D}^n$, then $T_f$ is compact if and only if $F=0$ a.e. with respect to a measure $\gamma$ associated with $\vartheta$ on the boundary $\partial\mathbb{D}^n$ . We also discuss the commuting problem: if $g$ is a non-constant bounded holomorphic function on $\mathbb{D}^n$, then what conditions does a bounded function $f$ need to satisfy so that $T_f$ commutes with $T_g$?