论文标题
在海森伯格组中的固有Lipschitz图上的Riesz Tranform
The Riesz tranform on intrinsic Lipschitz graphs in the Heisenberg group
论文作者
论文摘要
我们证明，海森伯格·里斯（Heisenberg Riesz）的变换为$ l_2 $ - 在第一个海森伯格集团$ \ mathbb {h} $的一系列固有的lipschitz图中无限。我们通过将\ cite {ny2}的方法与停止时间的论点相结合，并通过引入几种新技术来分析内在Lipschitz图的奇异积分来构建这个家庭。这些包括一个riesz转换的公式，该公式是根据垂直平面上的奇异积分和界面上的奇异积分的界限，这是由图引起的奇异积分的流动。在途中，我们使用我们的施工表明，强烈的几何引理在$ \ mathbb {h} $中均为$ [2,4）$的所有指数。 Our results are in stark contrast to two fundamental results in Euclidean harmonic analysis and geometric measure theory: Lipschitz graphs in $\mathbb{R}^n$ satisfy the strong geometric lemma, and the $m$--Riesz transform is $L_2$--bounded on $m$--dimensional Lipschitz graphs in $\mathbb{R}^n$ for $m\in （0，n）$。
We prove that the Heisenberg Riesz transform is $L_2$--unbounded on a family of intrinsic Lipschitz graphs in the first Heisenberg group $\mathbb{H}$. We construct this family by combining a method from \cite{NY2} with a stopping time argument, and we establish the $L_2$--unboundedness of the Riesz transform by introducing several new techniques to analyze singular integrals on intrinsic Lipschitz graphs. These include a formula for the Riesz transform in terms of a singular integral on a vertical plane and bounds on the flow of singular integrals that arises from a perturbation of a graph. On the way, we use our construction to show that the strong geometric lemma fails in $\mathbb{H}$ for all exponents in $[2,4)$. Our results are in stark contrast to two fundamental results in Euclidean harmonic analysis and geometric measure theory: Lipschitz graphs in $\mathbb{R}^n$ satisfy the strong geometric lemma, and the $m$--Riesz transform is $L_2$--bounded on $m$--dimensional Lipschitz graphs in $\mathbb{R}^n$ for $m\in (0,n)$.
