ESTIMATES FOR $\bar\partial$ AND HANKEL OPERATORS ON GENERALIZED FOCK SPACES ON ${\mathbb C}^n$

Abstract

Let $\varphi: \mathbb{C}^n \to\mathbb{R}$ be a $C^2$ plurisubharmonic function on $\mathbb{C}^n$. Suppose that there exist $C_1, C_2 \gt 0$ such that $\sup_{\mathbb{C}^n} |\bar\partial \partial\varphi| \lt C_1$ and $H_{\varphi}(\xi,\xi)(z) \geq C_2 |\xi|^2$ for $\xi \in \mathbb{R}^{2n}$ and $z \in \mathbb{C}^n$, where $H_{\varphi}(\xi,\xi)(z)$ is the real Hessian of $\varphi$ at $z$. We prove $L^{p, \varphi}$ estimates for $\bar\partial$ on $\mathbb{C}^n$ for all $p \in [1,\infty]$. Moreover, by using the estimates for $\bar\partial$, we characterize boundedness and compactness of Hankel operators with anti-holomorphic symbols on generalized Fock spaces on $\mathbb{C}^n$.