Cauchy Integral, Fractional Sobolev Spaces and Chord-Arc Curves

TL;DR

This paper explores fractional Sobolev spaces associated with Jordan curves, examining conditions for their equivalence and implications for boundary value problems like the Plemelj-Calderón problem.

Contribution

It characterizes when two fractional Sobolev spaces coincide on Jordan curves and analyzes the geometric conditions affecting this equivalence, extending classical results.

Findings

Equality of spaces holds for Lipschitz curves.

Chord-arc property is necessary and sufficient for $s=1/2$.

For general $s$, the equivalence depends on curve regularity.

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. ......