H\"older estimates for the $\bar\partial$ problem for $(p,q)$ forms on product domains

Abstract

The purpose of this paper is to study H\"older 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 H\"older 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 $C^{k, \alpha'}$ solution to the $\bar\partial$ problem on product domains for any $0<\alpha'<\alpha$ with the desired H\"older estimate.