An optimal decay estimate for the linearized water wave equation in 2D (Q2821735)

In this article the author proves a decay estimate for solutions of the 1D linear dispersive equation NEWLINE\[NEWLINEi\partial_tu=\partial_{xx}^{1/4}u,\;\;\text{for}\;\;(t,x)\in{\mathbb R}\times{\mathbb R}.NEWLINE\]NEWLINE He finds a decay \(|t|^{-1/2}\) when the initial data \(\phi=u(0)\) satisfies \(\phi\in H^1({\mathbb R})\) and \(x\partial_x\phi\in L^2_x({\mathbb R})\).NEWLINENEWLINEHe also extends the results to the equation NEWLINE\[NEWLINEi\partial_tu=\partial_{xx}^{\alpha}u,NEWLINE\]NEWLINE for a wide range of \(\alpha>0\).