On the weighted $\bar{\partial }$-Neumann problem on unbounded domains

Abstract

Let $\Omega$ be an unbounded, pseudoconvex domain in ${\mathbb{C}}^n$ and let $\phi$ be a ${\mathcal{C}}^2$-weight function plurisubharmonic on $\Omega$. We show both necessary and sufficient conditions for existence and compactness of a weighted $\bar{\partial }$-Neumann operator $N_{\phi }$ on the space $L_{(0,1)}^2(\Omega ,e^{-\phi })$ in terms of the eigenvalues of the complex Hessian $({\partial }^2\phi /\partial z_j\partial {\bar{z}}_k{)}_{j,k}$ of the weight. We also give some applications to the unweighted $\bar{\partial }$-Neumann problem on unbounded domains.

References in corpus (1)

• Pointwise estimates for the Bergman kernel of the weighted Fock space

Cited by in corpus (4)

• An idea on proving weighted Sobolev embeddings
• Closed range of $\bar{\partial }$ in $L^2$-Sobolev spaces on unbounded domains in ${\mathbb{C}}^n$
• Closed range of $\bar{\partial }$ on unbounded domains in ${\mathbb{C}}^n$
• A remark on boundary estimates on unbounded $Z(q)$ domains in ${\mathbb{C}}^n$