Compactness of commutators of Toeplitz operators on q-pseudoconvex domains

Abstract

Let $\Omega$ be a bounded q-pseudoconvex domain in $\mathbb{C}^n$, $n \geq 2$ and let $1 \leq q \leq n-1$. If $\Omega$ is smooth, we find sufficient conditions for the $\overline\partial$-Neumann operator to be compact. If $\Omega$ is non-smooth and if $q \leq p \leq n-1$, we show that compactness of the $\overline\partial$-Neumann operator, $N_{p+1}$, on square integrable (0, p+1)-forms is equivalent to compactness of the commutators $[B_p,\overline z_j]$, $1 \leq j \leq n$, on square integrable $\overline\partial$-closed (0, p)-forms, where $B_p$ is the Bergman projection on (0, p)-forms. Moreover, we prove that compactness of the commutator of $B_p$ with bounded functions percolates up in the $\overline\partial$-complex on $\overline\partial$-closed forms and square integrable holomorphic forms. Furthermore, we find a characterization of compactness of the canonical solution operator, $S_{p+1}$, of the $\overline\partial$-equation restricted on (0, p+1)-forms with holomorphic coefficients in terms of compactness of commutators $[T_p^{z_j*},T_p^{z_j}]$, $1 \leq j \leq n$, on (0, p)-forms with holomorphic coefficients, where $T_p^{z_j}$ is the Bergman-Toeplitz operator acting on (0, p)-forms with symbol $z_j$. This extends to domains which are not necessarily pseudoconvex.

Keywords

• \overline\partial
• and
• \overline\partial-Neumann operator
• Bergman-Toeplitz operator
• q-convex domains