# 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

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

• I think your norm $\dot{H}^{1-\delta}$ should instead be $\dot{H}^{\frac{d-1}{2}-\delta}$. Jul 20, 2019 at 4:04
• Sure, I was thinking in three dimension. I edited Jul 22, 2019 at 18:35

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}