最终高斯局部树上的Riesz变换
Riesz transform on eventually Gaussian local trees
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中文总结 AI 辅助
研究均匀局部树上的Riesz变换，在体积增长和热核估计条件下，证明局部Dini条件保证L^p有界性，并确定交替Vicsek分形折叠上不等式成立的精确p范围。
中文摘要 AI 辅助
我们研究均匀局部树上的Riesz变换 $\mathcal{R}=\partial(-\Delta)^{-\frac{1}{2}}$，这些度量测度空间是局部实树，其典范Dirichlet形式由沿骨架的弱导数构建。参考测度 $m$ 可能关于长度测度 $\nu$ 奇异，因此 $\mathcal{R}$ 的有界性理解为从 $L^{p}(m)$ 到 $L^{p}(\nu)$。与分形流形和电缆系统相反，扩散在小尺度上是次高斯的，在大尺度上是高斯的。在均匀体积增长、热核的双侧估计和热核的逐点梯度估计下，我们证明尺度函数上的局部Dini条件蕴含 $\mathcal{R}$ 在 $L^{p}$ 上对所有 $p\in[2,\infty)$ 有界，从而对每个 $p\in(1,2]$ 有逆Riesz不等式。反之，$\mathcal{R}$ 对某个 $p<2$ 的有界性，或对某个 $p>2$ 的逆Riesz不等式，迫使空间在小尺度上是一维的。我们证明调和函数的逆Hölder不等式产生梯度估计，并验证所有假设对于具有有界位移生成元的几何群作用的空间成立。作为应用，对于 $\mathbb{Z}^{d}$ 中的交替Vicsek分形折叠，我们确定了Riesz和逆Riesz不等式成立的精确 $p$ 范围。
英文摘要
We study the Riesz transform $\mathcal{R}=\partial(-Δ)^{-\frac{1}{2}}$ on uniform local trees, metric measure spaces that are locally real trees and whose canonical Dirichlet form is built from weak derivatives along the skeleton. The reference measure $m$ may be singular with respect to the length measure $ν$, so boundedness of $\mathcal R$ is understood from $L^{p}(m)$ to $L^{p}(ν)$. In contrast with fractal-like manifolds and cable systems, the diffusion is sub-Gaussian at small scales and Gaussian at large scales. Under uniform volume growth, two-sided heat kernel estimates and a pointwise gradient estimate for the heat kernel, we prove that a local Dini condition on the scale function implies boundedness of $\mathcal{R}$ on $L^{p}$ for every $p\in[2,\infty)$, and hence the reverse Riesz inequality for every $p\in(1,2]$. Conversely, boundedness of $\mathcal{R}$ for some $p<2$, or a reverse Riesz inequality for some $p>2$, forces the space to be one-dimensional at small scales. We show that a reverse Hölder inequality for harmonic functions yields the gradient estimate, and verify all hypotheses for spaces carrying a geometric group action whose generators have bounded displacement. As an application, for the alternating Vicsek fractafold in $\mathbb Z^{d}$ we determine the exact ranges of $p$ for which the Riesz and reverse Riesz inequalities hold.
