多线性 Fourier 乘子算子的有界性研究

Multilinear Fourier multiplier operators have been actively pursued recently due to their widely applications in harmonic analysis, PDE and pseudo-differential operators. The boundedness of multilinear Fourier multiplier operators has been proved to be a powerful tool in mathematics. Considerable attention was paid to this direction in the last few decades. In this area we first established a multilinear Mihlin-Hörmander multiplier theorem by using a new Sobolev space. Second, we got some weighted estimates for multilinear Fourier multiplier operators on weighted L^p spaces. Based on these works, we will generalize the existed results to some spaces ( Morrey space, weighted Hardy space, and so on) under more weaker smooth conditions. To achieve the main purpose Fourier transform estimates, weighted Littlewood-Paley theory and multilinear interpolation will be used. We will also make use of the boundedness properties of vector-valued multilinear Calderón-Zygmund singular integral operators to study vector-valued multilinear Fourer multiplier operators. So multilinear Calderón-Zygmund singular integral operators with kernels under more weaker conditions are also considered in this project.

多线性 Fourier 乘子算子在调和分析、偏微分方程、拟微分算子等方面有着广泛的应用。多线性 Fourier 乘子问题现在已成为很多专家所关注的热点之一。2012年申请人通过引入一类新的 Sobolev 空间建立了多线性 Mihlin-Hörmander 乘子定理；2013年申请人又得到了多线性 Fourier 乘子算子在加权 L^p 空间中的某些有界性。本项目研究乘子光滑性条件降低后多线性 Fourier 乘子算子在某些函数空间（Morrey 空间, 加权 Hardy 空间等）中的有界性，并研究多线性极大 Fourier 乘子算子的有界性。为此我们拟采用加权的 Littlewood-Paley 理论、Fourier 变换估计、多线性插值等技术来建立这些有界性。本项目还借助于核函数光滑性较弱的向量值多线性 Calderón-Zygmund 算子研究向量值多线性 Fourier 乘子算子。

多线性 Fourier 乘子算子在调和分析、偏微分方程、拟微分算子等方面有着广泛的应用。多线性 Fourier 乘子问题现在已成为很多专家所关注的热点之一。本项目研究了乘子光滑性条件降低后多线性 Fourier 乘子算子在 L^p 空间中的加权有界性。为此我们采用加权的 Littlewood-Paley 理论、Fourier 变换估计等技术来建立这些有界性。借助于核函数光滑性较弱的向量值多线性 Calderón-Zygmund 算子的有界性建立了向量值多线性 Fourier 乘子算子的加权有界性。此外，还得到了一些核光滑性较弱的多线性 Littlewood-Paley 型算子的加权有界性结果。