On the Existence of Invariant Tori in Nearly-Integrable Hamiltonian Systems With Finitely Differentiable Perturbations

We prove the existence of invariant tori in Hamiltonian systems, which are analytic and integrable except a $2n$-times continuously differentiable perturbation ($n$ denotes the number of the degrees of freedom), provided that the moduli of continuity of the $2n$-th partial derivatives of the perturbation satisfy a condition of finiteness (condition on an integral), which is more general than a Hölder condition. So far the existence of invariant tori could be proven only under the condition that the $2n$-th partial derivatives of the perturbation are Hölder continuous.