On the Schatten class membership of Hankel operators on the unit ball

Abstract

A well-known theorem of K. Zhu \cite{6} asserts that, for $2 \leq p <\infty $, the Hankel operators $H_f$ and $H_{\bar f}$ on the Bergman space $L^2_a(B_n,dV)$ of the unit ball belong to the Schatten class ${\mathcal{C}}_p$ if and only if the mean oscillation $\MO(f)(z) = \{\widetilde{|f|^2}(z) - |\tilde f(z)|^2\}^{1/2}$ belongs to $L^p(B_n,(1-|z|^2)^{-n-1}dV(z))$. It is well known that, for trivial reasons, this theorem cannot be extended to the case $p \leq 2n/(n+1)$. This paper fills the gap between $2n/(n+1)$ and 2. More precisely, we prove that, when $2n/(n+1) < p < 2$, the same theorem holds true.