Cumulative angular cancellation in Maz'ya's endpoint inequalities

Abstract

We establish domain criteria for critical homogeneous potential inequalities in the plane. Assuming signed plane cancellation, uniform absolute summability of frozen annular moments at each boundary centre implies the nonlinear endpoint estimate on arbitrary bounded open sets. For the Newton kernel and trace-free quadratic integrands, uniform boundedness of signed cumulative moments over all source centres and truncation radii is necessary and sufficient; testing all translates and dilates of one fixed radial unit-mass mollifier suffices. Tangent-cone cancellation on finitely cornered Dini domains follows as a geometric corollary. On one explicit strictly convex, non-Dini $C^1$ domain, the estimate holds for $2{\xi }_1{\xi }_2$ and fails for ${\xi }_1^2-{\xi }_2^2$ with $\log \log (1/\epsilon )$ divergence. A Laplacian-measure extension transfers density estimates to PDE inequalities on the Dini class and on strictly convex $C^1$ domains. We also classify quadratic cancellation on polygons. The signed necessary-and-sufficient criterion is restricted to quadratic integrands.

arXiv comment38 pages, 2 figures. Substantially revised and retitled. Added a boundary atomic-sum sufficient criterion and an exact quadratic cumulative criterion on arbitrary bounded planar open sets, together with contrasting endpoint estimates on one strictly convex non-Dini $C^1$ domain. Extended the Laplacian-measure construction to strictly convex $C^1$ domains