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


$$\Vert (P\_ju)(P\_kv)\Vert\_{L^2(I\times\mathbb{R}^d)}\leq C(\delta)\Vert P\_ju(t\_0)\Vert\_{\dot{H}^{-1/2+\delta}}\Vert P\_kv(t\_0)\Vert\_{\dot{H}^{\frac{d-1}{2}-\delta}},$$
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 $j,k$?**



Bilinear Strichartz estimates for the Schrodinger equation
I don't understand why you have the $\delta$. One has the following estimate (see, for instance, Theorem 4.18 in the Clay lecture notes "Nonlinear Schrodinger Equations at Critical Regularity" by Rowan Killip and Monica Visan):



> 
> **Bilinear Strichartz.** Let $j,k$ be integers such that $j\leq k$. Then
>  $$\|(e^{it\Delta}P\_j f)(e^{it\Delta}P\_k g)\|\_{L\_{t,x}^2(\mathbb{R}\times\mathbb{R}^d)} \lesssim 2^{\frac{j(d-1)}{2} -\frac{k}{2}} \|f\|\_{L^2(\mathbb{R}^d)}\|g\|\_{L^2(\mathbb{R}^d)}.$$
> 
> 
> 


Let$\tilde{P}\_j$ and $\tilde{P}\_k$ be "fattened" Littlewood-Paley projectors such that their symbols are identically one on the support of the symbol of $P\_j$ and $P\_k$, respectively, for all $j,k\in\mathbb{Z}$, so that in particular, $P\_j\tilde{P}\_j = P\_j$ and $P\_k\tilde{P}\_k=P\_k$. We then have as a consequence of Bilinear Strichartz that


$$\|(e^{it\Delta}P\_j f)(e^{it\Delta}P\_k g)\|\_{L\_{t,x}^2(\mathbb{R}\times\mathbb{R}^d)} \lesssim 2^{\frac{j(d-1)}{2}-\frac{k}{2}} \|\tilde{P}\_j f\|\_{L^2(\mathbb{R}^d)} \|\tilde{P}\_k g\|\_{L^2(\mathbb{R}^d)}.$$
By Bernstein's lemma,
\begin{align}
2^{\frac{j(d-1)}{2}}\|\tilde{P}\_j f\|\_{L^2(\mathbb{R}^d)} &\sim \|\tilde{P}\_j f\|\_{\dot{H}^{\frac{d-1}{2}}(\mathbb{R}^d)} \lesssim\|f\|\_{\dot{H}^{\frac{d-1}{2}}(\mathbb{R}^d)},\\
2^{-\frac{k}{2}}\|\tilde{P}\_k g\|\_{L^2(\mathbb{R}^d)} &\sim \|\tilde{P}\_kg\|\_{\dot{H}^{-1/2}(\mathbb{R}^d)} \lesssim \|g\|\_{\dot{H}^{-1/2}(\mathbb{R}^d)}.
\end{align}


