The $C^{p'}$-regularity conjecture near $p=2$

Se-Chan Lee; Taehun Lee

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arXiv · 2609.09966

The $C^{p'}$-regularity conjecture near $p=2$

Se-Chan Lee·Taehun Lee

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Abstract

We prove the $C^{p'}$-regularity conjecture in every dimension when $p>2$ is sufficiently close to $2$. To this end, we establish improved H\"older estimates for the gradients of $p$-harmonic functions. These estimates also determine the first-order asymptotics of the optimal gradient H\"older exponent in every dimension. The proof combines compactness, harmonic rigidity of the limiting profiles, and a sharp uniform gap estimate for the first variation of the gradient excess.

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Se-Chan Lee, Taehun Lee. 2026-09-09. The $C^{p'}$-regularity conjecture near $p=2$. https://arxiv.org/abs/2609.09966

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