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Continuing the research initiated in \cite{Fr-Ki2}, we study the existence ofsolutions and their regularity for the cohomological equations $X u=f$ forlocally Hamiltonian flows (determined by the vector field $X$) on a compactsurface $M$ of genus $g\geq 1$. We move beyond the case studied so far by Forniin \cite{Fo1,Fo3}, when the flow is minimal over the entire surface and thefunction $f$ satisfies some Sobolev regularity conditions. We deal with theflow 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 toquantify optimal regularity of solutions. For this purpose we construct afamily of invariant distributions $\mathfrak{F}_{\bar t}$,$\bar{t}\in\mathscr{TF}^*$ that play the roles of the Forni's invariantdistributions introduced in \cite{Fo1,Fo3} by using the language of translationsurfaces. The distributions $\mathfrak{F}_{\bar t}$ are global in nature (asemphasized 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 todetermine the optimal regularity of the solutions for the cohomologicalequation, see Theorem 1.1 and 1.2. As a by-product, we also obtained,interesting in itself, a spectral result (Theorem 1.3) for theKontsevich-Zorich cocycle acting on functional spaces arising naturally at thetransition to the first-return map.

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