论文标题
紧凑型谎言组上的伪伪差异操作员和傅立叶积分运算符
Subelliptic pseudo-differential operators and Fourier integral operators on compact Lie groups
论文作者
论文摘要
在本回忆录中，我们将全局伪差异操作员的理论扩展到紧凑型谎言组上任意次摩恩尼亚式结构的设置。更准确地说，给定一个紧凑的谎言组$ g $，以及与向量字段的系统$ x = \ {x_1，\ cdots，x_k \} $相关的次毛质{l} $，我们引入了（subelliptic）pseudo-diffircer yclus ytrus to y to y y y y y y y y y y y y y y y \ \ cdots，x_1 \ cdots \ cdots，x_k \} $ [138]中开发的基质值定量过程。该理论将如下开发。首先，我们将研究该积分的奇异核，估计$ l^p $ - $ l^p $，$ h^1 $ -l^1 $，$ l^\ infty $ - $ bmo $ type type untermess typerness of thehörmanderhörmander类。在获得的估计之间，我们证明了著名的Sharp Fefferman $ l^p $ -Theorem和Calderón-Vaillancourt Theorem的典型版本。所获得的估计值将用于在sieliptic sobolev和besov空间上建立下层算子的界限。我们将研究开发的下骨积分的椭圆度，参数的构造，热迹线和痕迹的正则化。将建立一个亚细胞全球函数演算以及Hulanicki定理的下层次版本。还将应用在表征我们的下hörmander类中（证明这些类别独立于某些参数的定义）来确定的方法，以表征[90]中开发的全局Hörmander类。
In this memoir we extend the theory of global pseudo-differential operators to the setting of arbitrary sub-Riemannian structures on a compact Lie group. More precisely, given a compact Lie group $G$, and the sub-Laplacian $\mathcal{L}$ associated to a system of vector fields $X=\{X_1,\cdots,X_k\}$ satisfying the Hörmander condition, we introduce a (subelliptic) pseudo-differential calculus associated to $\mathcal{L},$ based on the matrix-valued quantisation process developed in [138]. This theory will be developed as follows. First, we will investigate the singular kernels of this calculus, estimates of $L^p$-$L^p$, $H^1$-$L^1$, $L^\infty$-$BMO$ type and also the weak (1,1) boundedness of these subelliptic Hörmander classes. Between the obtained estimates we prove subelliptic versions of the celebrated sharp Fefferman $L^p$-theorem and the Calderón-Vaillancourt theorem. The obtained estimates will be used to establish the boundedness of subelliptic operators on subelliptic Sobolev and Besov spaces. We will investigate the ellipticity, the construction of parametrices, the heat traces and the regularisation of traces for the developed subelliptic calculus. A subelliptic global functional calculus will be established as well as a subelliptic version of Hulanicki theorem. The approach established in characterising our subelliptic Hörmander classes (by proving that the definition of these classes is independent of certain parameters) will be also applied in order to characterise the global Hörmander classes on arbitrary graded Lie groups developed in [90].
