关键词: Hankel operators, boundedness, compactness, Schatten class, $\overline{\partial}$-equation, weighted Fock spaces

Abstract: Let $\phi$ be a smooth radial weight that decays faster than the class Gaussian ones. We obtain certain estimates for the reproducing kernels and the $L^p$-estimates for solutions of the $\overline{\partial}$-equation on the weighted Fock spaces $F_{\phi}^p~(1\leq p\leq\infty)$, which extends the classical Hörmander Theorem. Furthermore, for a suitable $f$, we completely characterize the boundedness and compactness of the Hankel operator $H_f:F_{\phi}^p\rightarrow L^q(\mathbb{C},{\rm e}^{-q\phi(\cdot)}{\rm d}m)$ for all possible $1\leq p,q<\infty$ and also characterize the Schatten-$p$ class Hankel operator $H_f$ from $F_{\phi}^2$ to $L^2(\mathbb{C},{\rm e}^{-2\phi}{\rm d}m)$ for all $0<p<\infty$. As an application, we give a complete characterization of the simultaneously bounded, compact and Schatten-$p$ classes Hankel operators $H_f$ and $H_{\overline{f}}$ on $F_\phi^2$.

Key words: Hankel operators, boundedness, compactness, Schatten class, $\overline{\partial}$-equation, weighted Fock spaces