Sobolev regularity of the canonical solutions to $\overline{\partial }$ on product domains

Zhang, Yuan

Comptes Rendus. Mathématique, Tome 362 (2024) no. G2, pp. 171-176

Let $\Omega$ be a product domain in ${ℂ}^n,n\ge 2$, where each slice has smooth boundary. We observe that the canonical solution operator for the $\overline{\partial }$ equation on $\Omega$ is bounded in $W^{k,p}\left(\Omega \right)$, $k\in {\mathbb{Z}}^{+},1<p<\infty$. This Sobolev regularity is sharp in view of Kerzman-type examples.

Classification : 32W05, 32A25, 32A36
Keywords: canonical solution, $\bar{\partial }$ equation, Bergman projection, product domains, Sobolev regularity

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