Cauchy Integral, Fractional Sobolev Spaces and Chord-Arc Curves
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913884670001152 |
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| author | Wei, Huaying Zinsmeister, Michel |
| author_facet | Wei, Huaying Zinsmeister, Michel |
| contents | Let $Γ$ be a bounded Jordan curve and $Ω_i,Ω_e$ its two complementary components. For $s\in(0,1)$ we define $\mathcal{H}^s(Γ)$ as the set of functions $f:Γ\to \mathbb C$ having harmonic extension $u$ in $Ω_i\cup Ω_e$ such that $$ \iint_{Ω_i\cup Ω_e} |\nabla u(z)|^2 d(z,Γ)^{1-2s} dxdy<+\infty.$$ If $Γ$ is further assumed to be rectifiable we define $H^s(Γ)$ as the space of measurable functions $f:Γ\to \mathbb C$ such that $$\iint_{Γ\times Γ}\frac{|f(z)-f(ζ)|^2}{|z-ζ|^{1+2s}} dσ(z)dσ(ζ)<+\infty.$$ When $Γ$ is the unit circle these two spaces coincide with the homogeneous fractional Sobolev space defined via Fourier series. For a general rectifiable curve these two spaces need not coincide and our first goal is to investigate the cases of equality: while the chord-arc property is the necessary and sufficient condition for equality in the classical case of $s=1/2$, this is no longer the case for general $s\in (0,1)$. We show however that equality holds for Lipschitz curves.
The second goal involves the Plemelj-Calderón problem. ...... |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04564 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cauchy Integral, Fractional Sobolev Spaces and Chord-Arc Curves Wei, Huaying Zinsmeister, Michel Complex Variables Classical Analysis and ODEs 42B20, 46E35, 31A05, 30H35 Let $Γ$ be a bounded Jordan curve and $Ω_i,Ω_e$ its two complementary components. For $s\in(0,1)$ we define $\mathcal{H}^s(Γ)$ as the set of functions $f:Γ\to \mathbb C$ having harmonic extension $u$ in $Ω_i\cup Ω_e$ such that $$ \iint_{Ω_i\cup Ω_e} |\nabla u(z)|^2 d(z,Γ)^{1-2s} dxdy<+\infty.$$ If $Γ$ is further assumed to be rectifiable we define $H^s(Γ)$ as the space of measurable functions $f:Γ\to \mathbb C$ such that $$\iint_{Γ\times Γ}\frac{|f(z)-f(ζ)|^2}{|z-ζ|^{1+2s}} dσ(z)dσ(ζ)<+\infty.$$ When $Γ$ is the unit circle these two spaces coincide with the homogeneous fractional Sobolev space defined via Fourier series. For a general rectifiable curve these two spaces need not coincide and our first goal is to investigate the cases of equality: while the chord-arc property is the necessary and sufficient condition for equality in the classical case of $s=1/2$, this is no longer the case for general $s\in (0,1)$. We show however that equality holds for Lipschitz curves. The second goal involves the Plemelj-Calderón problem. ...... |
| title | Cauchy Integral, Fractional Sobolev Spaces and Chord-Arc Curves |
| topic | Complex Variables Classical Analysis and ODEs 42B20, 46E35, 31A05, 30H35 |
| url | https://arxiv.org/abs/2506.04564 |