Tangent-cone cancellation and Maz'ya's Φ-inequalities on finitely cornered planar domains
Abstract
Let 0<α<2 and p=2/(2−α). We study critical nonlinear convolution inequalities for kernels K:ℝ2∖{0}→ℝm and functions Φ:ℝm→ℝ positively homogeneous of degrees α−2 and p. We assume angular Lipschitz regularity for Φ and angular C0,γ regularity for K, where 0<γ≤1 and γ=1 if p<2. On bounded finitely cornered planar domains with a common Dini tangent modulus on the boundary branches, we characterize |∫ΩΦ(K*f) dx|≤ C|f|L1p for real densities f∈ L1(ℝ2)∩ Lsloc(ℝ2), 1<s≤∞, by signed angular cancellation on the plane, tangent half-planes, and full vertex tangent cones. An analogous criterion holds on infinite sectors for compactly supported mean-zero densities. For the Newton kernel and each nonzero angularly Lipschitz, positively two-homogeneous Φ:ℝ2→ℝ with vanishing half-circle moments, we construct a bounded convex C1 domain ΩΦ, independent of ε, and smooth tests uε satisfying |Δuε|L1(ΩΦ)+|∂nuε|L1(∂ΩΦ)≤2, while, despite tangent cancellation, their nonlinear integrals grow in magnitude like loglog(1/ε). On the Dini class, a constant-preserving linear extension with compact support modulo the domain mean and controlled measure-valued Laplacian yields the same necessary-and-sufficient criterion for the Newton-kernel PDE inequality, without exact corner charts. We also construct infinite-dimensional cancellation spaces containing nonquadratic functions and classify the quadratic locus in polygon moduli.
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BibTeX
@misc{long2026tangentcone,
title = {{Tangent-cone cancellation and Maz'ya's $Φ$-inequalities on finitely cornered planar domains}},
author = {Zhouyu Long and Wenming Zou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.12755},
eprint = {2609.12755},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12755},
}
RIS
TY - UNPB
TI - Tangent-cone cancellation and Maz'ya's Φ-inequalities on finitely cornered planar domains
AU - Zhouyu Long
AU - Wenming Zou
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2609.12755
UR - https://arxiv.org/abs/2609.12755
AB - Let 0<α<2 and p=2/(2−α). We study critical nonlinear convolution inequalities for kernels K:ℝ^2∖{0}→ℝ^m and functions Φ:ℝ^m→ℝ positively homogeneous of degrees α−2 and p. We assume angular Lipschitz regularity for Φ and angular C^0,γ regularity for K, where 0<γ≤1 and γ=1 if p<2. On bounded finitely cornered planar domains with a common Dini tangent modulus on the boundary branches, we characterize |∫_ΩΦ(K*f) dx|≤ C|f|_L^1^p for real densities f∈ L^1(ℝ^2)∩ L^s_loc(ℝ^2), 1<s≤∞, by signed angular cancellation on the plane, tangent half-planes, and full vertex tangent cones. An analogous criterion holds on infinite sectors for compactly supported mean-zero densities. For the Newton kernel and each nonzero angularly Lipschitz, positively two-homogeneous Φ:ℝ^2→ℝ with vanishing half-circle moments, we construct a bounded convex C^1 domain Ω_Φ, independent of ε, and smooth tests u_ε satisfying |Δu_ε|_L^1(Ω_Φ)+|∂_nu_ε|_L^1(∂Ω_Φ)≤2, while, despite tangent cancellation, their nonlinear integrals grow in magnitude like loglog(1/ε). On the Dini class, a constant-preserving linear extension with compact support modulo the domain mean and controlled measure-valued Laplacian yields the same necessary-and-sufficient criterion for the Newton-kernel PDE inequality, without exact corner charts. We also construct infinite-dimensional cancellation spaces containing nonquadratic functions and classify the quadratic locus in polygon moduli.
ER -