AI 中文总结
研究具有逆平方势的薛定谔算子，从已知双边比较出发确定双权分数积分映射范围，给出特定条件下不等式成立情况，包括原点临界边界处的结果，还推导加权索伯列夫结果及处理哈代临界情形。
AI 中文摘要
设$H_a = -\Delta + a|x|^{-2}$是$L^2(\mathbb{R}^d)$上的弗里德里希斯扩张，其中$d\geq3$且$-(d - 2)^2/4\leq a\lt0$处于吸引性哈代范围。从$H_a^{-s/2}$核的已知正双边比较出发，确定了两个幂权的完整强非端点映射范围。若$\sigma=(d - 2 - \sqrt{(d - 2)^2 + 4a})/2$且$0\lt s\lt d - 2\sigma$，则在主定理所述的指数排序、缩放、求和及原点条件下，$1\lt p,q\lt\infty$时，[ ||x|^{-\beta}H_a^{-s/2}f|{L^q} \lesssim ||x|^\alpha f|{L^p} ]成立。在原点临界边界处，强估计及相应加权$L^p\to L^{q,\infty}$估计不成立，而洛伦兹替换$L^{p,1}\to L^{q,\infty}$成立。还推导了加权索伯列夫结果并单独处理了哈代临界弗里德里希斯情形。未声称有新的热核、谱乘子、伯恩斯坦或利特尔伍德 - 佩利定理。
英文摘要
Let $H_a=-Δ+a|x|^{-2}$ be the Friedrichs extension on $L^2(\mathbb{R}^d)$, where $d\ge 3$ and $-(d-2)^2/4\le a<0$ lies in the attractive Hardy range. Starting from the known positive two-sided comparison for the kernel of $H_a^{-s/2}$, we determine the complete strong non-endpoint mapping range for two power weights. If $σ=(d-2-\sqrt{(d-2)^2+4a})/2$ and $0<s<d-2σ$, then [ ||x|^{-β}H_a^{-s/2}f|{L^q} \lesssim ||x|^αf|{L^p} ] holds for $1<p,q<\infty$ precisely under the exponent ordering, scaling, sum, and origin conditions stated in the main theorem. At either origin-critical boundary, the strong estimate and the corresponding weighted $L^p\to L^{q,\infty}$ estimate fail, whereas the Lorentz replacement $L^{p,1}\to L^{q,\infty}$ holds. We also derive weighted Sobolev consequences and treat the Hardy-critical Friedrichs case separately. No new heat-kernel, spectral multiplier, Bernstein, or Littlewood--Paley theorem is claimed.
Comments8 pages, no figures
