Work

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

Year: 2026

Type: preprint

Abstract: We settle two endpoint problems for Beurling--Carleson support conditions. For $03$ 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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2608.08715

Authors: Fang, Pengcheng, He, Yixin

Institutions: Shanghai University of Finance and Economics, Fudan University

Topic: Advanced Harmonic Analysis Research

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

OpenAlex ID: W7202183170

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