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

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 $C^{k,α}$ components, $0\le p\le n, 1\le q\le n$, $k\in \mathbb Z^+\cup \{0\}, 0<α\le 1$, there is a $C^{k, α'}$ solution to the $\bar\partial$ problem on product domains for any $0<α'<α$ with the desired Hölder estimate.

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