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

Abstract

We settle two endpoint problems for Beurling--Carleson support conditions. For $0<s<\frac{1}{2}$ and $\theta =\frac{1-2s}{1-s}$, we construct a nonatomic singular probability measure $\mu$, supported on a single $\theta$-Beurling--Carleson set, such that $S_{\nu }^{\prime }\notin H^s$ for every nonzero submeasure $0<\nu \le \mu$. For $m>3$ 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.

17 pages, no figures