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

Abstract

We settle two endpoint problems for Beurling--Carleson support conditions. For $0<s<\frac12$ 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 $S_ν&#39;\notin H^s$ 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.