Sharp Holder regularity for autonomous Beltrami equations

Jie Huang; Jian-Feng Zhu

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arXiv · 2610.25702

Sharp Holder regularity for autonomous Beltrami equations

Jie Huang·Jian-Feng Zhu

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Abstract

Let $Ω\subset\mathbb C$ be a domain and let $\mathcal H:\mathbb C\to\mathbb C$ be Lipschitz with constant $0<k<1$. We prove that every weak $W^{1,2}_{\mathrm{loc}}$ solution of the autonomous Beltrami equation $f_{\bar z}=\mathcal H(f_z)$ belongs to $C^{1,γ}_{\mathrm{loc}}$, where $γ=3/(2K+1)$ and $K=(1+k)/(1-k)$. The exponent is optimal, resolving the conjecture of Astala, Clop, Faraco, Jääskeläinen and Koski.

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Jie Huang, Jian-Feng Zhu. 2026-10-19. Sharp Holder regularity for autonomous Beltrami equations. https://arxiv.org/abs/2610.25702

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