2023/06/04 by Krzysztof Frączek, Frączek, Krzysztof, Minsung Kim +1
Mathematics · #37A10 #37C40 #37C83 #37E35 #37J12 #Algebraic Geometry and Number Theory #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2306.02340
openalex publication_date 2023/06/04 · openalex created_date 2023/06/07 · openalex updated_date 2026/07/28
Continuing the research initiated in \citeFr-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≥ 1. We move beyond the case studied so far by Forni in \citeFo1,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 \mathfrakF t, t∈\mathscrTF^* that play the roles of the Forni's invariant distributions introduced in \citeFo1,Fo3 by using the language of translation surfaces. The distributions \mathfrakF t are global in nature (as emphasized in the title of the article), unlike the distributions \mathfrakdkσ,j, (σ,k,j)∈\mathscrTD and \mathfrakCkσ,l, (σ,k,l)∈\mathscrTC introduced in \citeFr-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.