Fractional Sobolev Spaces on Quasicircles

Abstract: Let $Γ$ be a bounded Jordan curve and $Ωi,Ω_e$ its two complementary components. For $1<p<\infty,\,s\in(0,1)$ we define $\mathcal{B}{p,p}^s(Ω_{i,e})$ as the set of functions $f:Γ\to \mathbb C$ having harmonic extension $u$ respectively in $Ωi$ and $Ω_e$ such that $$ \iint{Ω{i,e}} |\nabla u(z)|^p d(z,Γ)^{(1-s)p-1} dxdy<+\infty.$$ If $Γ$ is further assumed to be rectifiable we define $B{p,p}^s(Γ)$ as the space of functions $f\in L^p(Γ)$ such that $$\iint_{Γ\times Γ}\frac{|f(z)-f(ζ)|^p}{|z-ζ|^{1+ps}} |dz||dζ|<+\infty.$$ When $Γ$ is the unit circle these three spaces coincide with the homogeneous fractional Besov-Sobolev space. For a general rectifiable curve these 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 $s=1/p,\, p\ge 2$, this is no longer the case for general $s\in (0,1)$. We show however that equality holds for radial-Lipschitz curves. In the general (possibly non-rectifiable) case we study boundary values of functions in $\mathcal{B}{p,p}^s(Ω{i,e})$ and give conditions for equality of these trace-spaces that we then call $\mathcal{B}^s_{p,p}(Γ)$. Using Plemelj-Calderón property we further identify $\mathcal{B}^s_{p,p}(Γ)$ with the space of restrictions of a weighted Sobolev space of the plane. Finally we re-interpretate some of our results as the "almost"-Dirichlet principle in the spirit of Maz'ya.