Tangent-cone cancellation and Maz'ya's $\Phi$-inequalities on finitely cornered planar domains

Abstract

Let $0<\alpha <2$, $p=2/(2-\alpha )$, and let $K:{\mathbb{R}}^2\setminus \{0\}\to {\mathbb{R}}^m$ and $\Phi :{\mathbb{R}}^m\to \mathbb{R}$ be positively homogeneous of degrees $\alpha -2$ and $p$, with Lipschitz angular parts. For bounded finitely cornered piecewise-$C^{1,\beta }$ planar domains $\Omega$, we characterize the critical estimate $|{\int }_{\Omega }\Phi (K\ast f)dx|\le C_{\Omega ,K,\Phi }\Vert f{\Vert }_{L^1({\mathbb{R}}^2)}^p$. It holds if and only if signed angular cancellation holds on the plane, the tangent half-planes, and the complete vertex cones. We obtain the analogous criterion on infinite sectors for compactly supported mean-zero densities. For domains with a finite exact ambient conformal-sector atlas, we construct a constant-preserving linear extension $E_{\Omega }$ whose Laplacian is a finite signed Radon measure controlled by $\Vert \Delta u{\Vert }_{L^1(\Omega )}+\Vert {\partial }_{n}u{\Vert }_{L^1(\partial \Omega )}$. For the Newton kernel this yields a necessary-and-sufficient tangent-model criterion for the corresponding Maz'ya $\Phi$-inequality. We also classify the quadratic cancellation locus in polygon moduli. At a genuine corner, the full vertex cone therefore carries an additional cancellation obstruction not detected by its incident tangent half-planes.

52 pages, 3 figures