Hölder estimates for the $\bar{\partial }$ problem for $(p,q)$ forms on product domains

Abstract

The purpose of this paper is to study Hölder estimates for the $\bar{\partial }$ problem for $(p,q)$ forms on products of general planar domains. As indicated by an example of Stein and Kerzman, solutions to the $\bar{\partial }$ problem on product domains in ${\mathbb{C}}^n(n\ge 2)$ does not gain regularity in Hölder spaces. Making use of an integral representation of Nijenhuis and Woolf, we show that given a $\bar{\partial }$-closed $(p,q)$ form with $C^{k,\alpha }$ components, $0\le p\le n,1\le q\le n$, $k\in {\mathbb{Z}}^{+}\cup \{0\},0<\alpha \le 1$, there is a $C^{k,{\alpha }^{\prime }}$ solution to the $\bar{\partial }$ problem on product domains for any $0<{\alpha }^{\prime }<\alpha$ with the desired Hölder estimate.

22 pages. To appear in International Journal of Mathematics