Low Regularity Solutions for the General Quasilinear Ultrahyperbolic Schrödinger Equation

Abstract

We present a novel method for establishing large data local well-posedness in low regularity Sobolev spaces for general quasilinear Schrödinger equations with non-degenerate and nontrapping metrics. Our result represents a definitive improvement over the landmark results of Kenig, Ponce, Rolvung and Vega (Kenig et al. in Adv Math 196:373–486, 2005; Carlos et al. in Adv Math 206:402–433, 2006; Carlos et al. in Invent Math 134:489–545, 1998; Carlos et al. in Invent Math 158:343–388, 2004), as it weakens the regularity and decay assumptions to the same scale of spaces considered by Marzuola, Metcalfe and Tataru in (Jeremy et al. in Arch Ration Mech Anal 242:1119-1175, 2021), but removes the uniform ellipticity assumption on the metric from their result. Our method has the additional benefit of being relatively simple but also very robust. In particular, it only relies on the use of pseudodifferential calculus for classical symbols.