Solving the cohomological equation for locally Hamiltonian flows, part II: Global obstructions

Abstract Continuing the research initiated in Adv. Math . ( 446 (2024), 109668), we provide a solution to the problem of existence and regularity of solutions to the cohomological equation for locally Hamiltonian flows (determined by a vector field ) on a compact surface of genus , when the flow is restricted to its minimal component. We go beyond the case studied so far by Forni in Ann. of Math . (2) ( 146 (1997), 295–344) and Ergodic Theory Dynam. Systems ( 41 (2021), 685–789), when the flow is minimal over the entire surface and the function satisfies some Sobolev regularity conditions. We deal with the flow restricted to any of its minimal components and any smooth function whenever the restricted flow satisfies the Full Filtration Diophantine Condition, a full measure condition. The main goal of this article is to quantify the optimal regularity of solutions. For this purpose, we construct a family of invariant distributions , , which play a role analogous to Forni's invariant distributions constructed in Ann. of Math . (2) ( 146 (1997), 295–344) and Ergodic Theory Dynam. Systems ( 41 (2021), 685–789) by using the language of translation surfaces. Unlike the locally defined distributions , , and , , introduced in Frączek and Kim ( Adv. Math . 446 (2024), 109668), the distributions are global in nature, as emphasized in the title of this article. All three families of distributions are used to determine the optimal regularity of solutions to the cohomological equation, see Theorems 1.2 and 1.3. As a by‐product, we also obtain an intrinsically interesting spectral result (Theorem 1.4) for the Kontsevich–Zorich cocycle acting on functional spaces that arise naturally at the transition to a first‐return map.