Sharp and Endpoint Two-Weight Fractional Integral Estimates for Schr"odinger Operators with Inverse-Square Potentials

Abstract

Let $H_a=-\hat{I}”+a|x{|}^{-2}$ 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 $\ddot{I}ƒ=(d-2-\sqrt{(d-2{)}^2+4a})/2$ and $0<s<d-2\ddot{I}ƒ$, 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.

8 pages, no figures