海森堡群中的低维一致可求长性——第2部分
On low-dimensional uniform rectifiability in Heisenberg groups - Part 2
- University of Jyväskylä（于韦斯屈莱大学）
- Università di Trento（特伦托大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该文在海森堡群中证明k维内禀Lipschitz图的几何引理，扩展了相关稳定性结果，为k正则集的corona分解建立必要条件，完善了低维一致可求长性的研究。
AI中文摘要：
设1≤k≤n，我们证明海森堡群$\boldsymbol{\text{H}}^n$中的k维内禀Lipschitz图满足关于水平β数的几何引理$\boldsymbol{\text{GLem}}(\beta_{2,\boldsymbol{\text{V}}_k},p)$，其中指数$p=p(k)$。此前该结果仅在k=1的情形下成立，我们的证明在该情形下恢复了尖锐指数p=4。对于k>1，我们采用Orponen最初为欧氏和抛物型Lipschitz函数开发的积分几何方法。此外，对于k=n，我们利用Morrey型不等式，展示如何直接从$\boldsymbol{\text{R}}^{2n}$中的各向同性Dorronsoro定理推导出几何引理。基于新的几何引理，我们建立了$\boldsymbol{\text{H}}^n$中k正则集允许内禀Lipschitz图的 corona 分解的必要条件，该条件已由最后两位作者与Pinamonti的早期工作证明是充分的，它涉及除$\beta_{2,\boldsymbol{\text{V}}_k}$之外的额外平坦系数。因此，在此过程中，我们将几何引理在“大碎片”函子下的已知稳定性结果扩展到更大类的系数。
英文摘要：
Let $1\leq k\leq n$. We prove that $k$-dimensional intrinsic Lipschitz graphs in the Heisenberg group $\mathbb{H}^n$ satisfy a geometric lemma $\mathrm{GLem}(β_{2,\mathcal{V}_k},p)$ for horizontal $β$-numbers with an exponent $p=p(k)$. Previously, this result was known only in the case $k=1$; our proof recovers the sharp exponent $p=4$ in this setting. For $k>1$, we adapt an integral geometric approach originally developed by Orponen for Euclidean and parabolic Lipschitz functions. In addition, for $k=n$, we show how to deduce a geometric lemma with $p=4$ directly from an isotropic Dorronsoro theorem in $\mathbb{R}^{2n}$ using a Poincaré inequality. Building on the new geometric lemmas, we establish a necessary condition for $k$-regular sets in $\mathbb{H}^n$ to admit corona decompositions by intrinsic Lipschitz graphs. The condition is known to be sufficient by earlier work of the last two authors together with Pinamonti. It involves additional flatness coefficients besides $β_{2,\mathcal{V}_k}$. Along the way, we therefore extend the known stability results for geometric lemmas under the ``big pieces'' functor to a larger class of coefficients.
