论文标题
关于双线性波浪互相相互作用的注释
A note on bilinear wave-Schrödinger interactions
论文作者
论文摘要
We consider bilinear restriction estimates for wave-Schrödinger interactions and provided a sharp condition to ensure that the product belongs to $L^q_t L^r_x$ in the full bilinear range $\frac{2}{q} + \frac{d+1}{r} < d+1$, $1 \leqslant q, r \leqslant 2$.此外，我们给出了一个反示例，表明即使在横向设置中，双线性限制估计也可能失败。该故障与缺乏锥体的曲率密切相关。最后，我们提到了这些估计值的扩展，以适应适用的功能空间。特别是，我们给出了$ u^2 $类型的一般转移类型原理，该原理大致暗示，如果估算均具有均质解决方案，则它也包含$ u^2 $。该转移参数可用于从均匀解决方案的相应边界中获得$ u^2 $中的双线性和多线性估计。
We consider bilinear restriction estimates for wave-Schrödinger interactions and provided a sharp condition to ensure that the product belongs to $L^q_t L^r_x$ in the full bilinear range $\frac{2}{q} + \frac{d+1}{r} < d+1$, $1 \leqslant q, r \leqslant 2$. Moreover, we give a counter-example which shows that the bilinear restriction estimate can fail, even in the transverse setting. This failure is closely related to the lack of curvature of the cone. Finally we mention extensions of these estimates to adapted function spaces. In particular we give a general transference type principle for $U^2$ type spaces that roughly implies that if an estimate holds for homogeneous solutions, then it also holds in $U^2$. This transference argument can be used to obtain bilinear and multilinear estimates in $U^2$ from the corresponding bounds for homogeneous solutions.
