A description of Poisson brackets on a space of nonlocal functionals (Q1070616)

The author introduces a ring \(\tilde F(\)J,k) (J index set, \(k={\mathbb{R}}\) or \({\mathbb{C}})\) of nonlocal functionals \(f: S^ J\to \{\phi |\) \(\phi '\in S\}=:S_{-1}\) (S: Schwartz space on \({\mathbb{R}})\), which is the extension of differential polynomials with coefficients in k by means of the operators T and \(\partial^{-1}:\) \[ T(u)=P.V.\frac{1}{2\delta}\int^{\infty}_{- \infty}ctg(\frac{\pi}{2\delta}(y-x))u(\quad y)dy,\quad \partial^{- 1}(u)=\int^{.}_{-\infty}u(y)-\int^{\infty}_{-\infty}u(y)dy, \] and the concept of Poisson brackets on \(\tilde F(\)J,k). Using this concept he can decribe the formally Hamiltonian operators that extend to Poisson brackets on \(\tilde F(\)J,k) and prove that the first and second Hamiltonian structures can be canonically extended to Poisson brackets on \(\tilde F(\)J,k).