In this survey we prove the sharpest results on the loss of Sobolev regularity for solutions of the cohomological equation for translation flows on translation surfaces, available to the methods developed by the author in Forni [Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus. Ann. of Math. (2)146(2) (1997), 295–344] and Forni [Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Ann. of Math. (2)155(1) (2002), 1–103]. The paper was mostly written between 2005 and 2006 while the author was at the University of Toronto, Canada, and was posted on arXiv in July 2007 [Forni. Sobolev regularity of solutions of the cohomological equation. Preprint, 2007, arXiv:0707.0940v2]. In an updated introduction we describe our results, taking into account later work on the problem and relevant recent progress in the field of Teichmüller dynamics, interval exchange transformations and translation flows.

中文翻译：

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上同调方程解的 Sobolev 正则性

在本次调查中，我们证明了对于平移表面上平移流的上同调方程解的 Sobolev 正则性损失的最尖锐结果，可用于作者在 Forni 中开发的方法 [Sobolev高等属的表面。安。数学。(2)146(2) (1997), 295–344] 和 Forni [高等属表面上面积保持流的遍历平均值的偏差。安。数学。(2)155(1) (2002), 1-103]。这篇论文主要写于 2005 年至 2006 年，当时作者在加拿大多伦多大学，并于 2007 年 7 月在 arXiv 上发表 [Forni. 上同调方程解的 Sobolev 正则性。预印本, 2007 年,arXiv:0707.0940v2]。在更新的介绍中，我们描述了我们的结果，考虑到后来在该问题上的工作以及 Teichmüller 动力学、区间交换转换和翻译流领域的相关最新进展。