Harmonic Analysis on Homogeneous Spaces and its Applications

研究了齐次李群空间上的傅立叶变换的性质。特别是，我们把注意力集中在李群的测不准原理上。首先，我们将经典傅里叶变换的Hardy定理推广到Cartan运动群。我们还建立了一般运动群的哈代定理的L p版，称为考林-普莱斯定理。其次，我们研究了半单李群的测不准原理。证明了连通非紧半单李群的Hardy定理的一个类似结果。这个定理可以用与群的连续主级数相对应的傅里叶变换来表述。我们还成功地证明了黎曼对称空间上向量丛的Cowling-Price定理和连通非紧半单李群的Cowling-Price定理。

We studied the properties of the Fourier transforms on homogeneous spaces of Lie groups. Especially, we focussed our attention on the the uncertainty principle for Lie groups. First, we generalized the Hardy theorem for the classical Fourier transform to the Cartan motion groups. We also established an L^p-version of the Hardy theorem, which is called the Cowling-Price theorem, for general motion groups.Next we studied the uncertainty principle for semisimple Lie groups. We proved an analogue of the Hardy theorem for connected noncompact semisimple Lie groups. The theorem can be stated by the Fourier transforms corresponding to the continuous principal series of the group. We also succeeded to prove the Cowling-Price theorem for vector bundles over Riemannian symmtric spaces and for connected noncompact semisimple Lie groups.