Bilinear Strichartz estimates for the Schrodinger equation

Question

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 $j,k$?

Answer 1

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}