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

Abstract: Let $H=-Δ+V(|x|)$ be a nonnegative radial Schrödinger operator on $\mathbb{R}<sup>d$, $d\ge 2$, whose positive harmonic function satisfies $U(r)\simeq r<sup>{-σ_0}$ for $0&lt;r\le 1$ and $U(r)\simeq r<sup>{-σ_\infty}$ for $r\ge 1$, with $-d/2&lt;σ<em>0,σ</em>\infty&lt;d/2$. 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 $H<sup>{-s/2}$ admits the clean two-sided estimate $$K_s^H(x,y)\simeq |x-y|<sup>{s-d}U(|x|)U(|y|)/[U(|x|+|x-y|)U(|y|+|x-y|)]$$, namely $0&lt;s&lt;\min{d,d-2σ<em>0,d-2σ</em>\infty}$. In this range we give a complete necessary-and-sufficient classification of the broken-power estimate $$|w_{-β<em>0,-β</em>\infty}H^{{-s/2}f|_{L^{{q,v}}\lesssim}} |w_{α<em>0,α</em>\infty}f|_{L<sup>{p,u}}$$ for $1&lt;p,q&lt;\infty$ and $1\le u,v\le\infty$. The result covers signed ground-state exponents, the full range $q&lt;p$, all one-sided weight equalities, both scale equalities, and simultaneous endpoint corners. Scale equality is governed by $u\le v$, including when $q&lt;p$; 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.

• The paper establishes a rigorous framework to derive sharp Lorentz-space bounds for fractional powers of radial Schrödinger operators with inverse-square potentials.
• It introduces broken-power weight structures to capture two-sided asymptotics and employs spectral and heat kernel techniques to classify the operator's kernel estimates.
• The analysis provides necessary and sufficient criteria for weight estimates, illuminating endpoint behaviors critical for dispersive PDE and spectral theory.

Sharp Lorentz Estimates for Radial Schrödinger Operators with Inverse-Square Asymptotics

Problem Setting and Motivation

The paper introduces a rigorous framework for the study of fractional-power integral operators associated with radial Schrödinger operators of the form $H = -\Delta + V(|x|)$, where $d \geq 2$. The potential $V$ is assumed to have inverse-square behavior at both the origin and infinity, but not necessarily globally homogeneous. The ground-state harmonic function $U(r)$ exhibits a broken power regime, $U(r) \sim r^{-\sigma_0}$ for small $r$ ($0 < r \leq 1$), and $U(r) \sim r^{-\sigma_\infty}$ for large $r$ ($r \geq 1$), capturing the two-sided asymptotic structure.

Fractional integration in this context extends classical Hardy–Littlewood–Sobolev (HLS) and Stein–Weiss inequalities from the Euclidean setting to weighted, non-homogeneous operators, requiring new techniques beyond convolution kernel analysis. The main technical advance is the precise classification of Lorentz-space boundedness for the fractional powers $d \geq 2$0, with explicit broken-power weight structure and sharp kernel asymptotics.

Kernel Structure and Maximal Range

Central to the analysis is the explicit description of the kernel $d \geq 2$1 of $d \geq 2$2, derived via spectral representation and heat kernel estimates in the radial regime. Under the assumption that $d \geq 2$3 is an $d \geq 2$4 weight and using the machinery of Ishige–Kabeya–Ouhabaz, the paper establishes the uniform comparability

$d \geq 2$5

for $d \geq 2$6, valid exactly in the sharp open interval

$d \geq 2$7

Each boundary arises from distinct geometric or analytic mechanisms: $d \geq 2$8 is due to the singularity near the diagonal; $d \geq 2$9 from small-radius comparable blocks; $V$0 from large-radius tails. Saturation or logarithmic divergence occurs precisely at the boundaries, confirming both necessity and sufficiency for this range.

Broken-Power Lorentz Space Classification

The principal result is a complete necessary-and-sufficient characterization of Lorentz-space two-weight estimates associated with $V$1. The broken-power weights

$V$2

are used as input and output factors, to generalize classical power-weighted inequalities. The estimate

$V$3

is classified for all admissible $V$4, $V$5. The parameters are encoded in seven nonnegative 'margins' — a local margin $V$6, four one-sided margins ($V$7), and two scale margins ($V$8). The broken power structure induces a 'sandwich' condition

$V$9

and allows for scale equality $U(r)$0 even when $U(r)$1, provided Lorentz indexes satisfy $U(r)$2.

Fine-index endpoint behavior is captured:

• Input-critical power (equality in $U(r)$3 or $U(r)$4) requires $U(r)$5,
• Output-critical power (equality in $U(r)$6 or $U(r)$7) requires $U(r)$8,
• Scale equality ($U(r)$9 or $U(r) \sim r^{-\sigma_0}$0 or $U(r) \sim r^{-\sigma_0}$1) necessitates $U(r) \sim r^{-\sigma_0}$2,
• At a same-side power/scale corner, only $U(r) \sim r^{-\sigma_0}$3 is admissible.

The proof uses a nine-block decomposition across origin, transition, and infinity regimes, with precise operator-theoretic arguments for rank-one endpoints, geometric annular packets, truncated Riesz potentials, and triangular sequence operators. Notably, the Lorentz norm cannot be reduced to an $U(r) \sim r^{-\sigma_0}$4 sum of local pieces except in geometric packet cases.

Numerical Results and Strong Claims

• The maximal kernel range is demonstrated to be optimal, with uniform comparability breaking down and divergence arising at all boundaries.
• The Lorentz two-weight estimate is classified exactly and exhaustively, including endpoint behaviors and all critical equalities.
• Strong-space reductions ($U(r) \sim r^{-\sigma_0}$5, $U(r) \sim r^{-\sigma_0}$6) recover classical Sobolev embedding results, showing strict inequalities for all one-sided margins when $U(r) \sim r^{-\sigma_0}$7.
• The methodology is robust against sign changes of ground-state exponents ($U(r) \sim r^{-\sigma_0}$8), with necessary modifications in the margin ceilings.
• The paper provides explicit diagnostics for all cases where estimates fail, driven by local or geometric packet divergence.

Implications and Future Directions

The theoretical implications are significant for the harmonic analysis of operators with critical inverse-square asymptotics, especially in the presence of non-homogeneous or radial perturbations. Practically, the results apply to fractional integral estimates for Schrödinger operators arising in dispersive PDE, spectral theory, and potential analysis with radial symmetry and variable asymptotics.

The rigorous Lorentz-space characterizations open the possibility for further study in several directions:

• Extension to logarithmic ground-state branches or potentials without radial symmetry.
• Characterization at critical fractional orders ($U(r) \sim r^{-\sigma_0}$9 or $r$0) where the kernel saturates or logarithmic factors arise.
• Endpoint cases with $r$1 or $r$2.
• Exploration of non-radial and angular effects in the broken power weights.

The clarity and precision of the operator-level criteria for Lorentz estimates also have utility in numerical analysis and explicit computation of fractional powers, especially where scale and endpoint behaviors are delicate.

Conclusion

This paper establishes sharp, exhaustive Lorentz-space estimates for fractional powers of radial Schrödinger operators with inverse-square asymptotics, using an explicit clean kernel and broken-power weight structure. The seven-margin necessary-and-sufficient criteria, along with fine-index endpoint classifications, provide a comprehensive framework bridging classical potential theory with modern operator analysis in variable geometries. The theoretical structure is robust and offers a foundation for further advances in weighted harmonic analysis and operator theory on singular spaces.