# Bilinear Strichartz estimates for the Schrodinger equation

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 $$j,k$$?