Hankel operators in the Bergman space and Schatten $p$-classes: The case $1<p<2$

Abstract:

K. Zhu proved in Amer. J. Math. 113 (1991), 147-167, that, for $2 \leq p < \infty$, the Hankel operators $H_{f}$ and $H_{\bar f}$ on the Bergman space 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 $L^{p}(D,(1-|z|^{2})^{-2}dA(z))$. In this paper we prove that the same result also holds when $1 < p < 2$.