Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917456197451776 |
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| author | Cai, Yingying Tolsa, Xavier |
| author_facet | Cai, Yingying Tolsa, Xavier |
| contents | We provide quantitative estimates for the dimension drop of harmonic measure. We show that for a domain $Ω= \mathbb{R}^{n+1} \setminus E$ where $E$ is an $s$-Ahlfors regular compact set satisfying a uniform $L^2$-based non-flatness condition $β_2 \ge δ_0$, the dimension of its harmonic measure is strictly less than $s$ for $s \in (n - cδ_0^2, n]$. For planar domains, we establish an analogous quantitative threshold $s_0 = 1 - cδ_0^2$ under Azzam's uniform non-flatness condition $β_\infty + β_{\operatorname{hole}} \ge δ_0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_21047 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries Cai, Yingying Tolsa, Xavier Analysis of PDEs Classical Analysis and ODEs We provide quantitative estimates for the dimension drop of harmonic measure. We show that for a domain $Ω= \mathbb{R}^{n+1} \setminus E$ where $E$ is an $s$-Ahlfors regular compact set satisfying a uniform $L^2$-based non-flatness condition $β_2 \ge δ_0$, the dimension of its harmonic measure is strictly less than $s$ for $s \in (n - cδ_0^2, n]$. For planar domains, we establish an analogous quantitative threshold $s_0 = 1 - cδ_0^2$ under Azzam's uniform non-flatness condition $β_\infty + β_{\operatorname{hole}} \ge δ_0$. |
| title | Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries |
| topic | Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2604.21047 |