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

RESUMO

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 $|\int_\Omega \Phi(K*f)\,dx|\leq C_{\Omega,K,\Phi}\|f\|_{L^1(\mathbb{R}^2)}^p$. 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 $\|\Delta u\|_{L^1(\Omega)}+\|\partial_n u\|_{L^1(\partial\Omega)}$. 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.