Brezis’s Fourier-degree problem below $\alpha=1/3$

Brezis’s problem asks whether one universal Fourier summation method recovers the topological degree for every sufficiently regular circle map, including the critical exponent $\alpha=1/3$. Kahane and Brezis identified this threshold; the universal endpoint question was left open after failure was shown for a particular regularization.

Known results

• Kahane: the standard sine-regularized degree formula fails for some $f\in C^{0,1/3}(\mathbb{T};\mathbb{S}^{1})$.
• A positive degree formula is known at the critical Sobolev regularity $W^{1/3,3}(\mathbb{S}^{1};\mathbb{S}^{1})$.
• Cieszyński proved nonexistence of an admissible universal summation process for $0<\alpha<1/3$.

2026 endpoint claims

The manuscript at arXiv:2607.25862 claims that no admissible universal summation process recovers $\deg f$ for every $f\in C^{0,1/3}(\mathbb{T};\mathbb{S}^{1})$, implying failure for all $\alpha\leq1/3$. A separate manuscript proves only $\alpha<1/3$ and explicitly says the endpoint remains open; the endpoint claim therefore lacks independent validation.

Current status (as of July 2026): Failure below $\alpha=1/3$ is supported by recent manuscripts, but the endpoint $\alpha=1/3$ is only claimed solved negatively and remains disputed.