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

Abstract

© 2020 Author(s). 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.