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

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 $S_{\nu }^{\mathrm{\prime }}\mathrm{\notin }{H}^s$ 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 $\mathrm{\Delta }u=(u_{+}{)}^m$. 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.