Interior C^1,α regularity for infinity-harmonic functions

What OpenAI claims

Proves uniform interior C^1,α_d regularity for bounded infinity-harmonic functions in every dimension d ≥ 3, for a positive exponent depending only on dimension. The gradient's supremum norm and Hölder seminorm on the half unit ball are bounded by a dimension-dependent constant times the oscillation on the unit ball. The exponent is not explicit.

Paper

1. Uniform Interior C^1,α Estimates for Infinity-Harmonic Functions

   4 October 2026 · Preprint, not peer reviewed · PDF

   We prove a uniform interior C^1,α_d estimate for bounded infinity-harmonic functions in every dimension d ≥ 3. For some α_d ∈ (0,1/3] depending only on the dimension, the gradient's supremum norm and αd-Hölder seminorm on B1/2 are bounded by a dimension-dependent constant times the oscillation on B1. The exponent is not explicit, and no endpoint or boundary regularity claim is made. In particular, infinity-harmonic functions are locally C1 on every open subset of ℝd, for d ≥ 3.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.