Let $d\geq 2$, $I\subset\mathbb{R}$ a time interval and $t_0\in I$. Let $u,v$ be solutions to the free Schrodinger equation on $\mathbb{R}^d$, i.e. $(i\partial_t+\Delta)u=(i\partial_t+\Delta)v=0.$ Let us fix $j,k\in\mathbb{Z}$. For every $\delta\geq 0$, we have the following Bourgain-type bilinear estimate

$$\Vert (P_ju)(P_kv)\Vert_{L^2(I\times\mathbb{R}^d)}\leq C(\delta)\Vert P_ju(t_0)\Vert_{\dot{H}^{-1/2+\delta}}\Vert P_kv(t_0)\Vert_{\dot{H}^{1-\delta}},$$

where $P_j$, $P_k$ are the Paley-Littlewood projections.

My doubt is the following: Can we choose the constant $C(\delta)$ to be uniform in $j,k$?


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