Function theoretical applications of Lagrangian spectral invariants

Entov and Polterovich considered the concept of heaviness and superheaviness by the Oh-Schwarz spectral invariants. The Oh-Schwarz spectral invariants are defined in terms of the Hamiltonian Floer theory. In this paper, we define heaviness and superheaviness by spectral invariants defined in terms of the Lagrangian Floer theory and provide their applications. As one of them, we define a relative symplectic capacity which measures the existence of Hamiltonian chord between two disjoint Lagrangian submanifolds and provide an upper bound of it in a special case. We also provide applications to non-degeneracy of spectral norms, a relative energy capacity inequality, fragmentation norm and non-displaceability.