AI 中文总结
研究\(d\)维柱体上受特定系数扰动的狄拉克算子，通过证明加权\(L^2\)空间中的二次估计，得到里斯变换的齐次加藤平方根估计，并利用局部化和缩放得出一般流形上扰动狄拉克算子的局部二次估计。
AI 中文摘要
在\(d\)维柱体\(\mathcal{C}= \mathbb{R}^k\times N\)上，其中\(N\)是闭流形作为底且大尺度维度\(k\in[1,d)\)，我们证明了受有界、可测且增生系数扰动的狄拉克算子在加权\(L^2\)空间中的二次估计。这尤其给出了与二阶散度形式椭圆算子相关的里斯变换在\(\mathcal{C}\)上的齐次加藤平方根估计，其系数可测且退化由穆肯霍普特\(A_2\)权控制。通过局部化和缩放，还得到了具有局部薄柱体几何且可能零内射半径的一般流形上扰动狄拉克算子的局部二次估计。
英文摘要
On $d$-dimensional cylinders $\mathcal{C}= \mathbb{R}^k\times N$, with a closed manifold $N$ as base and large scale dimension $k\in[1,d)$, we prove quadratic estimates in weighted $L^2$ space for Dirac operators perturbed by bounded, measurable and accretive coefficients. This gives in particular homogeneous Kato square root estimates on $\mathcal{C}$ for Riesz transforms associated with second order divergence form elliptic operators, having measurable coefficients with degeneracy governed by a Muckenhoupt $A_2$ weight. By localisation and scaling, it also yields local quadratic estimates for perturbed Dirac operators on general manifolds with locally thin cylindrical geometry, and possibly with zero injectivity radius.
Comments33 pages, 2 figures. Proposition 2.9 replaces former Proposition 2.6 (Poincaré inequality for vertical gradient fields) to fix an error. Proposition 3.11 has also been adapted
