无反射Riesz测度的刚性及Lipschitz图上Riesz变换的Sobolev估计
Rigidity of reflectionless Riesz measures and Sobolev estimates for Riesz transforms on Lipschitz graphs
- Beijing Institute of Technology（北京理工大学）
- City University of Hong Kong（香港城市大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对Xavier Tolsa提出的正无反射测度刚性猜想，在多项式增长和单点大尺度下质量界条件下，证明了两类平坦性结果，并建立了Lipschitz图上Riesz变换的Sobolev估计。
AI中文摘要：
本文研究了Xavier Tolsa在文献[16]中提出的正无反射测度刚性问题的相关课题。设$1\leq n<d$，我们考虑$\mathbb R^d$中$n$维Riesz变换的正无反射测度。在多项式上增长以及关于某一点在大尺度上的下质量界条件下，我们获得了无反射测度的两个平坦性结果。第一个结果涉及全局定义的Lipschitz图测度，该测度具有有限的$\dot H^{1/2}$能量，且其参数密度有界、一致为正，并且该密度对常数的偏离属于$L^2$。第二个情形涉及一个无反射测度，该测度在某个紧集外部与平坦测度一致。在这两种情形下，该测度都是$n$维仿射平面上的$n$维Hausdorff测度的常数倍。对于整数$s>n/2$，我们进一步获得了$H^s$估计，该估计在斜率与密度扰动属于$L^\infty$且足够小的条件下，用归一化Riesz场控制图梯度和密度扰动。这些常数与支撑集无关。
英文摘要：
In this paper, we focus on the positive reflectionless measures rigidity problem proposed by Xavier Tolsa in [16]. Let $1\leq n<d$, we consider positive reflectionless measures for the $n$-dimensional Riesz transform in $\mathbb R^d$. Under polynomial upper growth and a lower mass bound at large scales about one point, we obtain two flatness results for reflectionless measures. The first concerns globally defined Lipschitz graph measures with finite $\dot H^{1/2}$ energy and bounded, uniformly positive parameter densities whose deviations from a constant belong to $L^2$. The second case concerns a reflectionless measure that coincides with a flat measure outside a compact set. In both cases, the measure is a constant multiple of the $n$-dimensional Hausdorff measure on an affine $n$-plane. For integers $s>n/2$, we further obtain $H^s$ estimates controlling the graph gradient and density perturbation by the normalized Riesz field under smallness of the slope and density perturbation in $L^\infty$. The constants are independent of the supports.
