AI 中文总结
针对 $d\geq4$ 的离散薛定谔算子，通过费米面表面测度的傅里叶衰减界建立 $\ell^p$ 到 $\ell^{p'}$ 的一致预解估计，确定不同维数下的尖锐范围并推导相关衍生估计。
AI 中文摘要
设 $H_0$ 是 $\mathbb Z^d$（$d\geq4$）上的标准离散拉普拉斯算子，记 $R_0(z)=(H_0-z)^{-1}$，$p'$ 为 $p$ 的赫尔德共轭（当 $p=1$ 时 $p'=\infty$）。我们通过证明晶格色散关系费米面的表面测度的局部傅里叶衰减界，建立从 $\ell^p(\mathbb Z^d)$ 到 $\ell^{p'}(\mathbb Z^d)$ 的一致对角预解估计。在正则点，衰减由二次项消失的坐标方向数决定：经局部变量变换后，这些方向产生一维三次相位，其余方向为二次相位。对于 $d\geq5$，我们证明了尖锐全局范围 $1\leq p\leq\frac{2(d+2)}{d+5}$；在阈值能量之外，该范围在奇数维仍保持尖锐，而对于偶数维 $d\geq6$，尖锐范围改进为 $1\leq p\leq\frac{2(2d+5)}{2d+11}$。在四维情形，我们得到 $1\leq p<4/3$ 的一致估计，以及带有平方根对数损失的 $\ell^{4/3}$-to-$\ell^4$ 端点估计；匹配的各向异性克纳普型构造给出了对应的必要条件。我们还推导了 Birman--Schwinger 和 Kato 光滑估计、复特征值估计，以及改进的薄谱投影估计。
英文摘要
Let $H_0$ be the standard discrete Laplacian on $\mathbb Z^d$, $d\geq4$, let $R_0(z)=(H_0-z)^{-1}$, and let $p'$ denote the Hölder conjugate of $p$, with $p'=\infty$ when $p=1$. We establish uniform diagonal resolvent estimates from $\ell^p(\mathbb Z^d)$ to $\ell^{p'}(\mathbb Z^d)$ by proving local Fourier-decay bounds for surface measure on the Fermi surfaces of the lattice dispersion relation. At a regular point, the decay is governed by the number of coordinate directions in which the quadratic term vanishes: after a local change of variables, these directions produce cubic one-dimensional phases, while the remaining directions are quadratic. For $d\geq5$, we prove the sharp global range \[ 1\leq p\leq\frac{2(d+2)}{d+5}. \] Away from the threshold energies, this range remains sharp in odd dimensions, whereas for even $d\geq6$ the sharp range improves to \[ 1\leq p\leq\frac{2(2d+5)}{2d+11}. \] In dimension four, we obtain uniform estimates for $1\leq p<4/3$ and an $\ell^{4/3}$-to-$\ell^4$ endpoint estimate with a square-root logarithmic loss. Matching anisotropic Knapp-type constructions give the corresponding necessary conditions. We also derive Birman--Schwinger and Kato smoothing bounds, complex-eigenvalue estimates, and improved thin spectral projection estimates.
