Sobolev Inequalities and Ellipticity of Planar Linear Hamiltonian Systems

In this paper we will establish two different classes of ellipticity criteria, called the L-p criteria and the L-p-L-q criteria respectively, for planar linear Hamiltonian systems with periodic coefficients. The criteria are explicitly expressed using the L-p and L-q norms of coefficients and some known Sobolev constants. These results can be considered as the extensions of the famous Lyapunov stability criterion for Hill's equations.