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arXiv · 2608.08715

Beurling--Carleson Endpoint Counterexamples: Singular Inner Functions and a Semilinear Equation

Pengcheng Fang·Yixin He
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Abstract

We settle two endpoint problems for Beurling--Carleson support conditions. For $0 3$ and $\alpha=\frac{m-3}{m-1}$, we construct a nonatomic probability measure supported on a single $\alpha$-Beurling--Carleson set that is not the deficiency measure of any nearly maximal solution of $\Delta u=(u_+)^m$. 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.

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Pengcheng Fang, Yixin He. 2026-08-09. Beurling--Carleson Endpoint Counterexamples: Singular Inner Functions and a Semilinear Equation. https://arxiv.org/abs/2608.08715

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