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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?

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  • $\begingroup$ 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. $\endgroup$
    – Willie Wong
    Commented Apr 9, 2019 at 18:28

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