Pointwise Decay of solutions to the energy critical nonlinear Schrödinger equations

Abstract

In this note, we prove pointwise decay in time of solutions to the 3D energy-critical nonlinear Schrödinger equations assuming data in $L^1\cap H^3$. The main ingredients are the boundness of the Schrödinger propagators in Hardy space due to Miyachi \cite{Miyachi} and a fractional Leibniz rule in the Hardy space. We also extend the fractional chain rule to the Hardy space.

arXiv comment9 pages; In this new version, we extend the fractional chain rule to the Hardy space