On the high–low method for NLS on the hyperbolic space

Abstract

In this paper, we first prove that the cubic, defocusing nonlinear Schrödinger equation on the two dimensional hyperbolic space with radial initial data in Hs(H2) is globally well-posed and scatters when s > 3/4. Then, we extend the result to nonlinearities of order p > 3. The result is proved by extending the high–low method of Bourgain in the hyperbolic setting and by using a Morawetz type estimate proved by Staffilani and Ionescu.