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

Abstract: Let $H_a=-Δ+a|x|<sup>{-2}$ be the Friedrichs extension on $L<sup>2(\mathbb{R}<sup>d)$, where $d\ge 3$ and $-(d-2)^2/4\le a&lt;0$ lies in the attractive Hardy range. Starting from the known positive two-sided comparison for the kernel of $H_a<sup>{-s/2}$, we determine the complete strong non-endpoint mapping range for two power weights. If $σ=(d-2-\sqrt{(d-2)<sup>2+4a})/2$ and $0<s<d-2σ$, then [ ||x|^{-β}H_a^{-s/2}f|{L^q} \lesssim ||x|^αf|{L^p} ] holds for $1&lt;p,q&lt;\infty$ 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 $L^p\to L<sup>{q,\infty}$ estimate fail, whereas the Lorentz replacement $L^{p,1}\to L<sup>{q,\infty}$ 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.

• The paper rigorously characterizes when two‐weight Lp to Lq inequalities hold for fractional powers of Schrödinger operators with inverse‐square potentials under four precise conditions.
• It utilizes precise kernel estimates with a three-part Stein–Weiss decomposition to control competing singularities arising from the inverse‐square potential.
• The study also derives weighted Sobolev inequalities and endpoint Lorentz bounds, extending classical embeddings and mapping principles to singular Schrödinger contexts.

Sharp Two-Weight Fractional Integral Estimates for Inverse-Square Schrödinger Operators

Introduction and Problem Setting

The paper "Sharp and Endpoint Two-Weight Fractional Integral Estimates for Schrödinger Operators with Inverse-Square Potentials" (2607.09585) provides a rigorous classification of power-weighted (two-weight) $L^p \to L^q$ inequalities for fractional powers of the Schrödinger operator $H_a = -\Delta + a|x|^{-2}$ on $\mathbb{R}^d$ ($d \geq 3$), focusing on the attractive Hardy range $-\frac{(d-2)^2}{4} \leq a < 0$. The analysis is centered on mapping properties of the fractional integral operator $H_a^{-s/2}$, capturing the interplay between scaling, the singularity at the origin, and the sharp threshold behavior of weighted inequalities.

A key driver is a two-sided kernel comparison for $H_a^{-s/2}$ from Killip et al., characterizing the operator by a model integral kernel with three competing singular mechanisms: a classical Riesz potential term, and separate singularities at both the input and output origins. This structure generalizes the classical Stein--Weiss framework for Riesz potentials to the inverse-square context.

Main Results and Characterization

The main result is a comprehensive characterization of when the two-weight estimate

$$\bigl\| |x|^{-\beta} H_a^{-s/2} f \bigr\|_{L^q} \lesssim \bigl\| |x|^{\alpha} f \bigr\|_{L^p}$$

holds for $1 < p, q < \infty$ and weights $|x|^\alpha$ (input) and $H_a = -\Delta + a|x|^{-2}$0 (output). The exponents must satisfy four explicit conditions:

1. Exponent Ordering: $H_a = -\Delta + a|x|^{-2}$1
2. Scaling: $H_a = -\Delta + a|x|^{-2}$2
3. Sum Condition: $H_a = -\Delta + a|x|^{-2}$3
4. Origin Conditions: $H_a = -\Delta + a|x|^{-2}$4, $H_a = -\Delta + a|x|^{-2}$5

where

$H_a = -\Delta + a|x|^{-2}$6

and $H_a = -\Delta + a|x|^{-2}$7. The kernel does not allow for either the input or output origin condition to be attained as equality in the strong $H_a = -\Delta + a|x|^{-2}$8 estimate, providing strict boundaries at which the operator fails to be bounded.

At these critical boundaries, classical strong-type inequalities fail, but the authors show that the weak-type Lorentz analog holds: $H_a = -\Delta + a|x|^{-2}$9. Neither $\mathbb{R}^d$0 nor $\mathbb{R}^d$1 is valid at the boundary.

Additionally, a family of sharp weighted Sobolev inequalities for functions spectrally supported away from zero is derived, extending the mapping principle to weighted energy spaces, including $\mathbb{R}^d$2-Hardy--Sobolev embeddings for fractional powers.

Proof Techniques and Structural Decomposition

The analysis leverages precise kernel estimates for $\mathbb{R}^d$3, using the explicit model

$\mathbb{R}^d$4

to decompose the operator into three classical Stein--Weiss operators corresponding to different regions of integration, separated by the relative sizes of $\mathbb{R}^d$5, $\mathbb{R}^d$6, and $\mathbb{R}^d$7.

Sharp necessity is established by explicit concentration tests:

• Input-origin blowup via $\mathbb{R}^d$8 supported at the origin shows $\mathbb{R}^d$9 is necessary.
• Output-origin blowup via $d \geq 3$0 supported away from the origin shows $d \geq 3$1 is necessary.
• Scaling and sum conditions derive from homogeneity and translation arguments.

The sufficiency is handled by majorizing the model kernel by sums of appropriately weighted Riesz potentials, then applying the Stein--Weiss theorem to each and exploiting the disjoint support structure.

For the endpoint Lorentz theory, the argument utilizes precise norm estimates in Lorentz spaces and a detailed adaptation of the Hardy--Littlewood rearrangement inequality, demonstrating that at criticality, sharp off-diagonal Lorentz control is achievable despite the failure of strong type.

Endpoint and Hardy-Critical Cases

The Hardy-critical parameter $d \geq 3$2, corresponding to $d \geq 3$3, is treated separately. For $d \geq 3$4, the same quantitative characterizations hold. However, for certain classical choices (e.g., the Sobolev embedding for $d \geq 3$5, $d \geq 3$6), the parameter values hit the forbidden boundary for the origin condition, precluding strong-type estimates at the unweighted energy level.

The analysis demonstrates that such boundary phenomena are intrinsic and persist even in the presence of optimal kernel control, highlighting the distinction between the full-space and radial-invariant class (where the sum condition is weaker and misses obstruction by translation).

Implications and Theoretical Significance

These results provide the first explicit, sharp delineation of the mapping range for two-power-weighted fractional integrals associated with the full inverse-square Schrödinger operator, with rigorous endpoint Lorentz alternatives at the boundary. This addresses the gap between abstract Sawyer-type testing criteria and explicit power-weighted inequalities, and clarifies the effect of both the scaling invariance and the double singularity from the inverse-square potential.

The implications are substantial for the theory of weighted Sobolev spaces associated to singular potentials, nonlocal PDE with scaling-critical singularities, and for the further development of endpoint harmonic analysis connected to Schrödinger operators. The results also sharpen and extend the existing literature on Stein--Weiss inequalities, now incorporating fixed-origin singularities that arise naturally in quantum and dispersive settings.

Future directions include extension to $d \geq 3$7, $d \geq 3$8 endpoints, best constant analysis and extremizers, general Schrödinger potentials, and a more complete harmonic analysis (e.g., spectral multipliers or Littlewood--Paley theory) for $d \geq 3$9 beyond the inverse-square case.

Conclusion

This work achieves a complete and explicit characterization of sharp and endpoint two-weight inequalities for fractional integrals of Schrödinger operators with attractive inverse-square potentials. The results establish strict mapping ranges for strong-type inequalities, provide endpoint Lorentz analogs, and supply weighted Sobolev consequences, all derived via a precise reduction to classical harmonic analysis objects and optimal use of kernel decompositions. These findings clarify the landscape for weighted inequalities in the presence of scaling-critical and origin-singular structure, with implications for both applied and theoretical analysis involving singular Schrödinger operators (2607.09585).