Beurling–Carleson端点反例：奇异内函数与半线性方程
Beurling--Carleson Endpoint Counterexamples: Singular Inner Functions and a Semilinear Equation
中文总结 AI 辅助
该论文针对Beurling–Carleson支撑条件的两个端点问题构造反例，揭示了临界支撑条件的必要非充分性，证明结合了Cantor–Moran构造、核发散与非线性能量障碍。
中文摘要 AI 辅助
我们解决了Beurling–Carleson支撑条件下的两个端点问题：对于0<s<1/2且θ=(1−2s)/(1−s)，构造了一个非原子奇异概率测度μ，其支撑在单个θ-Beurling–Carleson集上，使得对每个非零子测度0<ν≤μ，都有S_ν'∉H^s；对于m>3且α=(m−3)/(m−1)，构造了一个支撑在单个α-Beurling–Carleson集上的非原子概率测度，该测度不是Δu=(u_+)^m的任何近极大解的亏格测度。由此，第一个问题中存在遗传失效，而第二个问题的临界支撑条件是必要但非充分的。证明结合了端点Cantor–Moran构造、核发散与非线性能量障碍。
英文摘要
We settle two endpoint problems for Beurling--Carleson support conditions. For $0<s<\frac12$ and $θ=\frac{1-2s}{1-s}$, we construct a nonatomic singular probability measure $μ$, supported on a single $θ$-Beurling--Carleson set, such that $S_ν'\notin H^s$ for every nonzero submeasure $0<ν\leqμ$. For $m>3$ and $α=\frac{m-3}{m-1}$, we construct a nonatomic probability measure supported on a single $α$-Beurling--Carleson set that is not the deficiency measure of any nearly maximal solution of $Δu=(u_+)^m$. Thus hereditary failure persists at the first endpoint, while the critical support condition in the second problem is necessary but not sufficient. The proofs combine endpoint Cantor--Moran constructions with kernel divergence and a nonlinear energy obstruction.
