Solving the cohomological equation for locally hamiltonian flows, part II -- global obstructions

Abstract

Continuing the research initiated in \cite{Fr-Ki2}, we study the existence of solutions and their regularity for the cohomological equations $X u=f$ for locally Hamiltonian flows (determined by the vector field $X$) on a compact surface $M$ of genus $g\geq 1$. We move beyond the case studied so far by Forni in \cite{Fo1,Fo3}, when the flow is minimal over the entire surface and the function $f$ satisfies some Sobolev regularity conditions. We deal with the flow restricted to any its minimal component and any smooth function $f$ whenever the flow satisfies the Full Filtration Diophantine Condition (FFDC) (this is a full measure condition). The main goal of this article is to quantify optimal regularity of solutions. For this purpose we construct a family of invariant distributions $\mathfrak{F}_{\bar t}$, $\bar{t}\in\mathscr{TF}^*$ that play the roles of the Forni's invariant distributions introduced in \cite{Fo1,Fo3} by using the language of translation surfaces. The distributions $\mathfrak{F}_{\bar t}$ are global in nature (as emphasized in the title of the article), unlike the distributions $\mathfrak{d}^k_{\sigma,j}$, $(\sigma,k,j)\in\mathscr{TD}$ and $\mathfrak{C}^k_{\sigma,l}$, $(\sigma,k,l)\in\mathscr{TC}$ introduced in \cite{Fr-Ki2}, which are defined locally. All three families are used to determine the optimal regularity of the solutions for the cohomological equation, see Theorem 1.1 and 1.2. As a by-product, we also obtained, interesting in itself, a spectral result (Theorem 1.3) for the Kontsevich-Zorich cocycle acting on functional spaces arising naturally at the transition to the first-return map.