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

What the math gives to ML

The paper identifies a positive fractional-integral kernel whose behavior differs from ordinary Riesz attention by an inverse-square singularity at both the query and source origins. The transferable asset is the explicit three-regime kernel: near-diagonal smoothing, distance-to-origin amplification, and a sharp integrability restriction controlled by the Hardy parameter. A practical neural adaptation is a geometry-aware attention or message-passing layer that replaces a generic distance kernel with this analytically motivated scale-invariant kernel, optionally learning the singularity strength and center.