Integrable Teichmüller spaces for analysis on Weil-Petersson curves
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916922809909248 |
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| author | Matsuzaki, Katsuhiko |
| author_facet | Matsuzaki, Katsuhiko |
| contents | The integrableTeichmüller space $T_p$ for $p \geq 1$ is defined by the $p$-integrability of Beltrami coefficients. We characterize a quasisymmetric homeomorphism $h$ in $T_p$ by the condition that $\log h'$ belongs to the real $p$-Besov space, with a certain modification applied in the case $p=1$. This is done as part of the arguments for establishing a biholomorphic correspondence $Λ$ from the product of $T_p$ for simultaneous uniformization of $p$-Weil-Petersson curves into the $p$-Besov space. In particular, this proves the real-analytic equivalence between $T_p$ and the real $p$-Besov space. Moreover, the Cauchy transform of Besov functions on Weil-Petersson curves can be expressed by the derivative of this holomorphic map $Λ$, and from this, the Calderón theorem in this setting is straightforward. It also follows that the Cauchy transforms on $p$-Weil-Petersson curves holomorphically depend on their embeddings as they vary in the Bers coordinates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20341 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integrable Teichmüller spaces for analysis on Weil-Petersson curves Matsuzaki, Katsuhiko Complex Variables The integrableTeichmüller space $T_p$ for $p \geq 1$ is defined by the $p$-integrability of Beltrami coefficients. We characterize a quasisymmetric homeomorphism $h$ in $T_p$ by the condition that $\log h'$ belongs to the real $p$-Besov space, with a certain modification applied in the case $p=1$. This is done as part of the arguments for establishing a biholomorphic correspondence $Λ$ from the product of $T_p$ for simultaneous uniformization of $p$-Weil-Petersson curves into the $p$-Besov space. In particular, this proves the real-analytic equivalence between $T_p$ and the real $p$-Besov space. Moreover, the Cauchy transform of Besov functions on Weil-Petersson curves can be expressed by the derivative of this holomorphic map $Λ$, and from this, the Calderón theorem in this setting is straightforward. It also follows that the Cauchy transforms on $p$-Weil-Petersson curves holomorphically depend on their embeddings as they vary in the Bers coordinates. |
| title | Integrable Teichmüller spaces for analysis on Weil-Petersson curves |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2508.20341 |