Mixed Poincaré and Fefferman--Phong inequalities for measure potentials on $2$-PI spaces

What the math gives to ML

The paper gives a constructive way to control an arbitrary positive measure potential by a first-order energy plus a reciprocal-scale multiplier, without assuming that the potential measure is doubling or absolutely continuous. This is transferable to neural fields and graph-based networks trained on highly singular, clustered, or unequal-weight sample measures: a potential term concentrated on difficult regions can be stabilized by an ambient smoothness energy and a locally computed critical-scale penalty. The most practical adaptation is a capacity-inspired Sobolev regularizer whose multiplier is determined by local sample mass rather than by a globally chosen weight decay. The expected benefit is improved stability and robustness when supervision or collocation points are concentrated on lower-dimensional sets.