论文标题
重新审视紧凑型谎言组上的奇异整体操作员
Oscillating singular integral operators on compact Lie groups revisited
论文作者
论文摘要
在[24，定理2']中，查尔斯·费弗曼（Charles Fefferman）证明了一类振荡的奇异积分的弱（1,1）界限，其中包括欧几里得laplacian $Δ的振荡光谱乘数， t_θ（-Δ）：=（1-δ）^{ - \ frac {nθ} {4}} e^{i（1-δ）^{\fracθ{2}}}}，\，0 \ leq leqθ<1。 \ end {equation}这项工作的目的是将Fefferman的结果扩展到在任何任意的紧凑型谎言组上振荡奇异积分。还考虑了在拉普拉斯 - 贝特拉米操作员的振荡光谱乘数上的应用。我们主要定理的证明说明了操作员内核的条件，其傅立叶变换（根据组的表示理论定义）与组的微层/几何特性之间的微妙关系。
In [24, Theorem 2'] Charles Fefferman has proved the weak (1,1) boundedness for a class of oscillating singular integrals that includes the oscillating spectral multipliers of the Euclidean Laplacian $Δ,$ namely, operators of the form \begin{equation} T_θ(-Δ):= (1-Δ)^{-\frac{nθ}{4}}e^{i (1-Δ)^{\fracθ{2}}},\,0\leq θ<1. \end{equation} The aim of this work is to extend Fefferman's result to oscillating singular integrals on any arbitrary compact Lie group. Applications to oscillating spectral multipliers of the Laplace-Beltrami operator are also considered. The proof of our main theorem illustrates the delicate relationship between the condition on the kernel of the operator, its Fourier transform (defined in terms of the representation theory of the group) and the microlocal/geometric properties of the group.
