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We obtain a decay estimate for solutions to the linear dispersive equation$iu_t-(-\Delta)^{1/4}u=0$ for $(t,x)\in\mathbb{R}\times\mathbb{R}$. Thiscorresponds to a factorization of the linearized water wave equation$u_{tt}+(-\Delta)^{1/2}u=0$. In particular, by making use of theLittlewood-Paley decomposition and stationary phase estimates, we obtain decayof 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 $\lVertx\partial_x\varphi\rVert_{L_x^2(\mathbb{R})}$. As another application of theseideas, we give an extension to equations of the form$iu_t-(-\Delta)^{\alpha/2}u=0$ for a wider range of $\alpha$.

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