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 $$ \|P_{>\lambda}u\bar{v}\|_{L^2_t(\mathbb R)L^2_x(\mathbb R)} \leq C \lambda^{-1/2} \|u_0\|_{L^2(\mathbb R)} \|v_0\|_{L^2(\mathbb R)} $$

Proof. See the Proposition 4.4 in this paper.

*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.

My question is: Is the higher dimension analogue of Proposition 4.4 is available? If so, can you provide reference? If not, what one expect?