The twisted cohomological equation over the partially hyperbolic flow

TL;DR

This paper investigates the twisted cohomological equation over higher-rank semisimple Lie groups, providing characterizations of obstructions, constructing smooth solutions, and establishing Sobolev estimates using novel representation theory techniques.

Contribution

It introduces the first comprehensive study of general twisted cohomological equations, employing Mackey theory and Mellin transform to analyze solutions and obstructions.

Findings

Characterization of obstructions to solving the equation

Construction of smooth solutions with Sobolev estimates

Development of new representation theory techniques

Abstract

Let $\mathbb{G}$ be a higher-rank connected semisimple Lie group with finite center and without compact factors. In any unitary representation $(\pi, \mathcal{H})$ of $\mathbb{G}$ without non-trivial $\mathbb{G}$-fixed vectors, we study the twisted cohomological equation $(X+m)f=g$, where $m\in\mathbb{R}$ and $X$ is in a $\mathbb{R}$-split Cartan subalgebra of $\text{Lie}(\mathbb{G})$. We characterize the obstructions to solving the cohomological equation, construct smooth solutions of the cohomological equation and obtain tame Sobolev estimates for $f$. We also study common solution to (the infinitesimal version of) the twisted cocycle equation $(X+m)g_1=(\mathfrak{v}+m_1)g_2$, where $\mathfrak{v}$ is nilpotent or in a $\mathbb{R}$-split Cartan subalgebra, $m,m_1\in\mathbb{R}$. This is the first paper studying general twisted equations. Compared to former papers, a new technique in representation theory is developed by Mackey theory and Mellin transform.