Contraction properties for holomorphic functions via isoperimetric stability on the Bergman ball
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arXiv
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| Format: | Preprint |
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2026
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| author | Kalaj, David Zhu, Jian-Feng |
| author_facet | Kalaj, David Zhu, Jian-Feng |
| contents | We prove a local contraction property for holomorphic functions that are nearly constant, relating weighted Bergman spaces $A^p_α(\B_n)$ and $A^q_β(\B_n)$. Our approach converts geometric information on weighted superlevel sets into analytic deficit inequalities and rests crucially on a quantitative stability result (of Fuglede type) for the isoperimetric inequality in the Bergman ball. As an application, along the contractive line $q/p=β/α$, we obtain a deficit contraction near the extremizer $f\equiv 1$: if $f=1+ϕ$ with $ϕ$ small and its weighted level sets are nearly spherical (after recentering), then the $A^q_β$-deficit is controlled by the $A^p_α$-deficit, and the same deficit quantitatively controls the deviation of the level sets from spheres. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_22524 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Contraction properties for holomorphic functions via isoperimetric stability on the Bergman ball Kalaj, David Zhu, Jian-Feng Complex Variables Functional Analysis We prove a local contraction property for holomorphic functions that are nearly constant, relating weighted Bergman spaces $A^p_α(\B_n)$ and $A^q_β(\B_n)$. Our approach converts geometric information on weighted superlevel sets into analytic deficit inequalities and rests crucially on a quantitative stability result (of Fuglede type) for the isoperimetric inequality in the Bergman ball. As an application, along the contractive line $q/p=β/α$, we obtain a deficit contraction near the extremizer $f\equiv 1$: if $f=1+ϕ$ with $ϕ$ small and its weighted level sets are nearly spherical (after recentering), then the $A^q_β$-deficit is controlled by the $A^p_α$-deficit, and the same deficit quantitatively controls the deviation of the level sets from spheres. |
| title | Contraction properties for holomorphic functions via isoperimetric stability on the Bergman ball |
| topic | Complex Variables Functional Analysis |
| url | https://arxiv.org/abs/2603.22524 |