The purpose of this paper is to study Hölder estimates for the ∂̄ problem for (p,q) forms on products of general planar domains. As indicated by an example of Stein and Kerzman, solutions to the ∂̄ problem on product domains in ℂn(n ≥ 2) does not gain regularity in Hölder spaces. Making use of an integral representation of Nijenhuis and Woolf, we show that given a ∂̄-closed (p,q) form with Ck,α components, 0 ≤ p ≤ n, 1 ≤ q ≤ n, k ∈ ℤ+ ∪{0}, 0 < α ≤ 1, there is a Ck,α′ solution to the ∂̄ problem on product domains for any 0 < α′ < α with the desired Hölder estimate.

中文翻译：

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产品域上 (p,q) 形式的 ∂̄ 问题的 Hölder 估计

本文的目的是研究 Hölder 估计∂̄问题为(p,q)在一般平面域的产品上形成。正如 Stein 和 Kerzman 的例子所示，解决方案∂̄产品领域的问题ℂn(n ≥ 2)在 Hölder 空间中没有规律性。利用 Nijenhuis 和 Woolf 的积分表示，我们证明给定一个∂̄-关闭(p,q)形成与Cķ,α成分，0 ≤ p ≤ n, 1 ≤ q ≤ n,ķ ∈ ℤ+ ∪{0}, 0 < α ≤ 1，有一个Cķ,α'的解决方案∂̄任何产品领域的问题0 < α' < α与所需的 Hölder 估计。