CR几何和共形几何有明显的区别和联系, 可以把共形几何上经典的定义和分析学推广到CR流形上, 但是CR几何有着更丰富的几何结构. CR流形平坦模型是Heisenberg群, 它的地位相当于共形几何中的欧几里得空间. 由于非局部的分数阶Laplace算子研究的兴起, 分数阶的偏微分方程的研究引起了关注. 2007年, Caffarell和Silvestre将关于非局部的分数阶Laplace算子的研究转化为对于局部的退化椭圆算子的研究. 虽然维数增加了一维, 但是椭圆方程的经典理论如正则性, 极值原理等都能够得到应用. 这些理论使得研究分数阶微分方程可以更多借鉴经典整数阶情形.在CR流形情形, Gonzalez-Qin等数学家从散射理论, 缠结算子等理论出发建立分数阶sub-Laplace算子, 以及其它相关概念和理论，如分数阶Q曲率.给定分数阶Q曲率问题是经典Yamabe问题的自然推广.

Recently the geometry of CR manifolds has attracted much attention. When the CR manifold is strictly pseudoconvex, there are many parallels with Riemannian geometry. The geometry is rich and the abstract models are real hypersurfaces in complex manifolds, particularly the Heisenberg group. The Heisenberg group plays the same role as $\mathbb{R}^n$ in conformal geometry and is CR isomorphism to the sphere $S^{2n+1}$ minus a point. A common strategy in CR geometry is analogy with conformal geometry. In the Caffarelli and Silvestre' remarkable paper, fractional order operators are treated as the Dirichlet-to-Neumann operator of a degenerate elliptic problem. On the CR setting, fractional CR covariant operators of order $2\gamma$, $\gamma\in \mathbb{R}$, may be defined from scattering theory by Gonz\'{a}lez-Qing. They also define the fractional Q-curvature. Some research of fractional Q-curvature problem on CR manifolds are from classical Yamabe problem.