Sharp Broken-Power Lorentz Estimates for Fractional Powers of Radial Schr\"odinger Operators with Inverse-Square Asymptotics

Abstract

Let $H=-\Delta+V(|x|)$ be a nonnegative radial Schr\"odinger operator on $\mathbb{R}^d$, $d\ge 2$, whose positive harmonic function satisfies $U(r)\simeq r^{-\sigma_0}$ for $0<r\le 1$ and $U(r)\simeq r^{-\sigma_\infty}$ for $r\ge 1$, with $-d/2<\sigma_0,\sigma_\infty<d/2$. 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 $0<s<\min\{d,d-2\sigma_0,d-2\sigma_\infty\}$. In this range we give a complete necessary-and-sufficient classification of the broken-power estimate $\|w_{-\beta_0,-\beta_\infty}H^{-s/2}f\|_{L^{q,v}}\lesssim \|w_{\alpha_0,\alpha_\infty}f\|_{L^{p,u}}$ 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.