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

We obtain a decay estimate for solutions to the linear dispersive equation $iu_t-(-\Delta)^{1/4}u=0$ for $(t,x)\in\mathbb{R}\times\mathbb{R}$. This corresponds to a factorization of the linearized water wave equation $u_{tt}+(-\Delta)^{1/2}u=0$. In particular, by making use of the Littlewood-Paley decomposition and stationary phase estimates, we obtain decay of order $|t|^{-1/2}$ for solutions corresponding to data $u(0)=\varphi$, assuming only bounds on $\lVert \varphi\rVert_{H_x^1(\mathbb{R})}$ and $\lVert x\partial_x\varphi\rVert_{L_x^2(\mathbb{R})}$. As another application of these ideas, we give an extension to equations of the form $iu_t-(-\Delta)^{\alpha/2}u=0$ for a wider range of $\alpha$.