Cauchy Integral, Fractional Sobolev Spaces and Chord-Arc Curves

Abstract

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. ......