$L^2$函数的多线性加权卷积及其在非线性色散方程中的应用
Multilinear weighted convolution of $L^2$ functions, and applications to non-linear dispersive equations
摘要：
由Beals，Bourgain，Kenig-Ponce-Vega，Klainerman-Machedon等人使用的$X^{s,b}$空间是研究非线性色散方程低正则性的基本工具。它是特别感兴趣的，以获得双线性或多线性估计涉及这些空间。通过Plancherel定理和对偶性，这些估计归结为估计一个加权卷积积分的分量函数的$L^2$规范。本文系统地研究了$L^2$上的加权卷积估计。因此，我们得到尖锐的双线性估计的KdV，波，和薛定谔$X^{s,b}$空间。
英文摘要：
The $X^{s,b}$ spaces, as used by Beals, Bourgain, Kenig-Ponce-Vega, Klainerman-Machedon and others, are fundamental tools to study the low-regularity behaviour of non-linear dispersive equations. It is of particular interest to obtain bilinear or multilinear estimates involving these spaces. By Plancherel's theorem and duality, these estimates reduce to estimating a weighted convolution integral in terms of the $L^2$ norms of the component functions. In this paper we systematically study weighted convolution estimates on $L^2$. As a consequence we obtain sharp bilinear estimates for the KdV, wave, and Schrödinger $X^{s,b}$ spaces.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
