The Palais\u2013Smale condition for the Hamiltonian action on a mixed regularity space of loops in cotangent bundles and applications

Abstract

We show that the Hamiltonian action satisfies the Palais–Smale condition over a “mixed regularity” space of loops in cotangent bundles, namely the space of loops with regularity H^s, sin (frac{1}{2}, 1), in the base and H^{1-s} in the fiber direction. As an application, we give a simplified proof of a theorem of Hofer–Viterbo on the existence of closed characteristic leaves for certain contact type hypersufaces in cotangent bundles.