Sharp constants for weak estimates of Riesz Potentials when $0<s<\min\{n,2\}$

What the math gives to ML

The paper supplies an explicit sharp $L^1\to L^{q,\infty}$ control for the fractional nonlocal operator $I_s$, with $q=n/(n-s)$ and $0<s<\min\{n,2\}$. The transferable asset is not merely a fractional convolution, but a principled normalization and tail guarantee: bounded input mass implies a dimension- and order-dependent bound on the measure of large responses. A practical neural analogue is a fixed or learnable Riesz aggregation layer for image or point-grid features, combined with weak-norm monitoring or regularization to prevent rare, spatially diffuse activation explosions.