Dunkl变换在数学和物理中有着重要的应用，它是经典Fourier变换的推广。Dunkl变换所基于的Dunkl核函数和Dunkl测度具有比经典指数函数和Lebesgue测度更复杂的构造，所以对Dunkl变换及相关对象的研究有一定难度。本文重点研究了与Dunkl变换相关的不确定性原理和Bochner-Riesz平均算子的有界性。不确定性原理是量子力学中的一个基本原理，反映了深刻的对偶关系；Bochner-Riesz猜想是调和分析中的四大猜想之一，与其相关的理论联系了数学中的多个分支。所以，对这两个主题的研究具有重要意义。
  
  具体地，我们首先研究了与Dunkl-Wigner变换相关的一些性质，比如时频转化公式、衰减性、反演公式等。当窗口函数是恒不为0的径向Schwartz函数时，我们证明了与Dunkl-Wigner变换相关的Lieb不等式和弱不确定性原理。其次，我们定义了一类新的原子Hardy空间$ H^1_G $，$ H^1_G $中的原子支集包含于球的轨道，区别于Coifman和Weiss齐型空间意义下的Hardy空间$H^1_d$中的原子。我们证明了不同$ L^q $范数意义下的原子Hardy空间$ H^1_G $是彼此等价的。我们还证明了$ H^1_G $和$ \text{BMO}_G $空间的对偶性，并通过对偶性比较了$ H^1_d $和$ H^1_G $的空间大小关系，特别地，当取有限反射群为$ \mathbb{Z}_2 $ 时，我们有$ H^1_{d,\mathbb{Z}_2} \subsetneq H^1_{\mathbb{Z_2}} $；我们通过建立Dunkl分子理论，证明了Dunkl-Bochner-Riesz算子在$ H^1_G $上的有界性。

Dunkl transform plays a crucial role in Mathematics and Physics. It is a generalization of classical Fourier transform. The Dunkl kernel function and Dunkl measure which Dunkl transform is based on are more complicated than the classical exponential function and Lebesgue measure. Hence, it is more difficult to study Dunkl transform and some other subjects concerning it. This thesis mainly focuses on the uncertainty principle and boundedness of Bochner-Riesz operator associated with Dunkl transform. Uncertainty principle is a fundamental law in quantum mechanics. It reflects the Duality in nature and mathematics. Bochner-Riesz conjecture is one of four well-known conjectures in harmonic analysis. Several branches in Mathematics have connection with it. Thus, there is great significance in the research of these two aspects.
    
     Specifically, we first study some properties of Dunkl-Wigner transform, such as time-frequency shift formula, decay property, inversion formula, etc. When the window function is a radial Schwartz function that is non-zero everywhere, we prove a Lieb inequality and weak type uncertainty principle associated with the Dunkl-Wigner transform. Secondly, we define a new type of atomic Hardy space $ H^1_G $ where the support of an atom is allowed to be contained in the orbit of a ball. This characteristic makes atoms in $ H^1_G $ distinct from those in $ H^1_d $ in the homogenous space of Coifman and Weiss. We prove that atomic Hardy spaces $ H^1_G $ whose atoms are defined in different $ L^q $ norms are equivalent to one another. We also show that the duality between $ H^1_G $ and $ \text{BMO}_G $ space. And by this dual relation, we compare the size of $ H^1_d $ with $ H^1_G $. In particular, if the Weyl group is $ \mathbb{Z}_2 $, we have $ H^1_{d,\mathbb{Z}_2} \subsetneq H^1_{\mathbb{Z_2}} $. By establishing Dunkl molecule theory, we prove that Dunkl-Bochner-Riesz operator has $ H^1_G $ boundedness.