具有逆平方渐近性的径向薛定谔算子分数幂的尖锐破幂洛伦兹估计
Sharp Broken-Power Lorentz Estimates for Fractional Powers of Radial Schrödinger Operators with Inverse-Square Asymptotics
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中文总结 AI 辅助
研究具有逆平方渐近性的径向薛定谔算子分数幂，在特定假设下确定\(H^{-s/2}\)核双边估计的区间，给出\(1 < p, q < \infty\)且\(1 \leq u, v \leq \infty\)时破幂估计的充要分类，涵盖多种情况，证明结合多种分析和定理。
中文摘要 AI 辅助
设\(H = -\Delta + V(|x|)\)是\(\mathbb{R}^d\)（\(d\geq2\)）上的非负径向薛定谔算子，其正调和函数满足在\(0 < r \leq 1\)时\(U(r) \simeq r^{-\sigma_0}\)，在\(r \geq 1\)时\(U(r) \simeq r^{-\sigma_\infty}\)，其中\(-d/2 < \sigma_0, \sigma_\infty < d/2\)。在Ishige、Kabeya和Ouhabaz的双边基态热核估计假设下，确定了\(H^{-s/2}\)的核允许干净双边估计\(K_s^H(x,y) \simeq |x - y|^{s - d}U(|x|)U(|y|)/[U(|x| + |x - y|)U(|y| + |x - y|)]\)的最大开区间，即\(0 < s < \min\{d, d - 2\sigma_0, d - 2\sigma_\infty\}\)。在此区间内，给出了\(1 < p, q < \infty\)且\(1 \leq u, v \leq \infty\)时破幂估计\(\|w_{-\beta_0,-\beta_\infty}H^{-s/2}f\|_{L^{q,v}} \lesssim \|w_{\alpha_0,\alpha_\infty}f\|_{L^{p,u}}\)的完整充要分类。结果涵盖有符号基态指数、\(q < p\)的全范围、所有单边权重等式、两个尺度等式以及同时的端点角。尺度等式由\(u \leq v\)控制，包括\(q < p\)时；输入或输出功率端点分别要求\(u = 1\)或\(v = \infty\)；在同侧功率/尺度角，唯一允许的对是\((u, v) = (1, \infty)\)。证明结合干净核分析、局部洛伦兹 - 哈代 - 利特伍德 - 索伯列夫估计、秩一端点论证、几何环形序列空间、三角矩阵定理和九块分解。
英文摘要
Let $H=-Δ+V(|x|)$ be a nonnegative radial Schrödinger operator on $\mathbb{R}^d$, $d\ge 2$, whose positive harmonic function satisfies $U(r)\simeq r^{-σ_0}$ for $0<r\le 1$ and $U(r)\simeq r^{-σ_\infty}$ for $r\ge 1$, with $-d/2<σ_0,σ_\infty<d/2$. Assuming the two-sided ground-state heat-kernel estimate of Ishige, Kabeya, and Ouhabaz, we determine the maximal open range in which the kernel of $H^{-s/2}$ admits the clean two-sided estimate $K_s^H(x,y)\simeq |x-y|^{s-d}U(|x|)U(|y|)/[U(|x|+|x-y|)U(|y|+|x-y|)]$, namely $0<s<\min\{d,d-2σ_0,d-2σ_\infty\}$. In this range we give a complete necessary-and-sufficient classification of the broken-power estimate $\|w_{-β_0,-β_\infty}H^{-s/2}f\|_{L^{q,v}}\lesssim \|w_{α_0,α_\infty}f\|_{L^{p,u}}$ for $1<p,q<\infty$ and $1\le u,v\le\infty$. The result covers signed ground-state exponents, the full range $q<p$, all one-sided weight equalities, both scale equalities, and simultaneous endpoint corners. Scale equality is governed by $u\le v$, including when $q<p$; an input or output power endpoint requires respectively $u=1$ or $v=\infty$; and at a same-side power/scale corner the only admissible pair is $(u,v)=(1,\infty)$. The proof combines clean-kernel analysis, local Lorentz-Hardy-Littlewood-Sobolev estimates, rank-one endpoint arguments, geometric annular sequence spaces, a triangular matrix theorem, and a nine-block decomposition.
