Sharp Broken-Power Lorentz Estimates for Fractional Powers of Radial Schrödinger Operators with Inverse-Square Asymptotics

Abstract

Let $H=-\Delta +V(|x|)$ be a nonnegative radial Schrödinger 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 $\Vert w_{-{\beta }_0,-{\beta }_{\infty }}{H}^{-s/2}f{\Vert }_{L^{q,v}}\lesssim \Vert w_{{\alpha }_0,{\alpha }_{\infty }}f{\Vert }_{L^{p,u}}$ 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.

26 pages, no figures