Cauchy Integral, Fractional Sobolev Spaces and Chord-Arc Curves

Abstract

Let $\Gamma$ be a bounded Jordan curve and $\Omega_i,\Omega_e$ its two complementary components. For $s\in(0,1)$ we define $\mathcal{H}^s(\Gamma)$ as the set of functions $f:\Gamma\to \mathbb C$ having harmonic extension $u$ in $\Omega_i\cup \Omega_e$ such that $$ \iint_{\Omega_i\cup \Omega_e} |\nabla u(z)|^2 d(z,\Gamma)^{1-2s} dxdy<+\infty.$$ If $\Gamma$ is further assumed to be rectifiable we define $H^s(\Gamma)$ as the space of measurable functions $f:\Gamma\to \mathbb C$ such that $$\iint_{\Gamma\times \Gamma}\frac{|f(z)-f(\zeta)|^2}{|z-\zeta|^{1+2s}} d\sigma(z)d\sigma(\zeta)<+\infty.$$ When $\Gamma$ 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-Calder\'on problem. ......