(untitled) (European Journal of Pure and Applied Mathematics, 2025)
After the fundamental work of Livschitz in L1,L2, various research directions emerged, among which the following stand out: (i) the study of cocycles with values in groups and semigroups beyond R, as well as the investigation of corresponding regularity results; (ii) the analysis of how a certain degree of regularity (C^k for k=1,2,…,∞ ,ω) of the cocycle can confer corresponding regularity to the solution of the cohomological equation; and (iii) the study of higher-dimensional cohomology naturally associated with the action of groups such as Z^k or R^k. The aim of this article is to present, as self-contained as possible, a review of the natural generalizations of the notions of cocycles and cochains, as well as their corresponding results, in the study of cohomological equations.