有限角平面域上的切锥消没与Maz'ya $\Phi$-不等式
Tangent-cone cancellation and Maz'ya's $Φ$-inequalities on finitely cornered planar domains
- Department of Mathematical Sciences, Tsinghua University（清华大学数学科学系）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文在有限角平面域上刻画Maz'ya $\Phi$-不等式的临界估计，证明其成立等价于平面、切半平面及顶点锥上的带符号角消没条件，并构造保常数线性延拓，揭示真实角点处存在额外消没障碍。
中文摘要 AI 辅助
设$0<\alpha<2$，$p=2/(2-\alpha)$，并设$K:\mathbb{R}^2\setminus\{0\}\to\mathbb{R}^m$与$\Phi:\mathbb{R}^m\to\mathbb{R}$分别为次数$\alpha-2$与$p$的正齐次函数，且具有Lipschitz角部。对于有界有限角分段$C^{1,\beta}$平面域$\Omega$，我们刻画临界估计$|\int_\Omega \Phi(K*f)\\,dx|\leq C_{\Omega,K,\Phi}\\|f\\|_{L^1(\mathbb{R}^2)}^p$。该估计成立当且仅当在平面、切半平面及完整顶点锥上满足带符号角消没条件。我们在无穷扇区上对紧支撑均值为零的密度获得了类似准则。对于具有有限精确环境共形扇区图册的域，我们构造了一个保常数的线性延拓$E_\Omega$，其Laplacian是一个由$\\|\Delta u\\|_{L^1(\Omega)}+\\|\partial_n u\\|_{L^1(\partial\Omega)}$控制的有限符号Radon测度。对于Newton核，这为相应的Maz'ya $\Phi$-不等式给出了一个必要且充分的切模型准则。我们还对多边形模空间中的二次消没轨迹进行了分类。在真正的角点处，完整顶点锥因此携带了其相邻切半平面未检测到的额外消没障碍。
英文摘要
Let $0<α<2$, $p=2/(2-α)$, and let $K:\mathbb{R}^2\setminus\{0\}\to\mathbb{R}^m$ and $Φ:\mathbb{R}^m\to\mathbb{R}$ be positively homogeneous of degrees $α-2$ and $p$, with Lipschitz angular parts. For bounded finitely cornered piecewise-$C^{1,β}$ planar domains $Ω$, we characterize the critical estimate $|\int_ΩΦ(K*f)\,dx|\leq C_{Ω,K,Φ}\|f\|_{L^1(\mathbb{R}^2)}^p$. 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 $E_Ω$ whose Laplacian is a finite signed Radon measure controlled by $\|Δu\|_{L^1(Ω)}+\|\partial_n u\|_{L^1(\partialΩ)}$. For the Newton kernel this yields a necessary-and-sufficient tangent-model criterion for the corresponding Maz'ya $Φ$-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.
