A new proof of some discrete inequalities with standard central-difference type operators

摘要

Some discrete inequalities such as the Sobolev inequality give useful a priori estimates for numericalschemes. Although they had been known for the simplest forward difference operator,those for central difference type opereators had been left open until quite recently in Kojima–Matsuo–Furihata (2016) a unified way to discuss them was found. Still, due to some technicalreasons, the result was limited to a narrow range of central difference operators. In this paper,we provide a new proof that gives a complete answer regarding the discrete Sobolev inequalityand the discrete Gagliardo–Nirenberg inequality with the nonlinear Schr¨odinger equationindex.