Consider the homogeneous Schrödinger equation: $$i\partial_t u + \Delta u=0, u(x,0)= u_0(x)$$
Proposition 4.4 Let $$\lambda>0$$. Assume that $$u, v$$ are solutions to the homogeneous Schrödinger equation. Then $$\|P_{>\lambda}u\bar{v}\|_{L^2_t(\mathbb R)L^2_x(\mathbb R)} \leq C \lambda^{-1/2} \|u_0\|_{L^2(\mathbb R)} \|v_0\|_{L^2(\mathbb R)}$$
Koch, Herbert; Tataru, Daniel, Energy and local energy bounds for the 1-d cubic NLS equation in $$H^{-\frac{1}{4}}$$, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 29, No. 6, 955-988 (2012). ZBL1280.35137.
• As stated it has the wrong scaling for higher dimensions. If my back of envelope quick calculation is right, the $\lambda^{-1/2}$ should be replaced by $\lambda^{d/2 - 1}$ for the equation on $\mathbb{R}^d$ to get the correct scaling; and the left hand side should be replaced by the low frequency projector instead of the high frequency one. I haven't checked however whether the inequality is true. – Willie Wong Apr 9 at 18:28