Poisson structures on weak Sobolev loop spaces and applications to integrable systems

Abstract: We develop a framework for Poisson geometry on loop spaces of low regularity, extending Mokhov's classical constructions from smooth loops to weak Sobolev spaces $W^{s,p}(\mathbb{S^1},\mathbb{R}^m)$ with $o < s \frac{1}{2}$ and $1 < p < \infty.$ Within this setting we construct presymplectic and Poisson structures of hydrodynamic type, as well as their weakly non local deformations involving inverse derivatives. The analytic backbone relies on the boundedness of fractional multipliers, Hilbert transforms, and Lipschitz Nemytski operators on $W^{s,p}$, which ensures that all operations used in Mokhov's formalisn remain well defined at this level of regularity. We further show that teh horizontal-vertical bicomplex underlying variational Poisson geometry can be extended to $W^{s,p},$ so that the cohomological arguments proving skew-symmetry and the Jacobi identity carry over verbatim. As an application, we embed the Hamiltonian formalisms of several integrable PDEs (KdV, nonlinear Schr\"odinger, Camassa--Holm, and hydrodynamic systems of Dubrovin--Novikov type) into this weak Sobolev setting. Local order-one Poisson operators and their weakly nonlocal extensions are shown to be well posed for $W^{s,p}$ loops, while higher-order operators (e.g. the second KdV bracket) require stronger regularity. Our results provide a rigorous analytic foundation for Poisson geometry on weak loop spaces and open the way for extending the Hamiltonian theory of integrable systems beyond the smooth category.