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

We obtain a decay estimate for solutions to the linear dispersive equation $i{u}_t-(-\mathrm{\Delta }{)}^{1\mathrm{/}4}u=0$ for $(t,x)\in \mathbb{R}\times \mathbb{R}$. This corresponds to a factorization of the linearized water wave equation $u_{tt}+(-\mathrm{\Delta }{)}^{1\mathrm{/}2}u=0$. In particular, by making use of the Littlewood-Paley decomposition and stationary phase estimates, we obtain decay of order $\mathrm{\mid }t{\mathrm{\mid }}^{-1\mathrm{/}2}$ for solutions corresponding to data $u(0)=\phi$, assuming only bounds on $\parallel \phi {\parallel }_{H_x^1(\mathbb{R})}$ 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 $i{u}_t-(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}u=0$ for a wider range of $\alpha$.