Fractional Sobolev Spaces on Quasicircles
Huaying Wei, Michel Zinsmeister
TL;DR
This work extends fractional Besov–Sobolev spaces from the unit circle to general quasicircles by introducing $\mathcal{B}_{p,p}^s(Ω)$ via weighted harmonic energy and defining trace spaces $\mathcal{B}_{p,p}^s(Γ)$. It develops a Plemelj–Calderón framework for decomposing boundary data into holomorphic parts on $Ω_i$ and $Ω_e$, using Beurling-type operators, conformal invariance, and $A_p$ weights; it also connects these trace spaces to weighted Sobolev spaces on the plane and establishes an almost-Dirichlet principle. The paper proves equality of trace spaces in several geometric regimes: chord-arc and radial-Lipschitz curves yield $\mathcal{B}_{p,p}^s(Ω\toΓ)= B_{p,p}^s(Γ)$ for relevant ranges of $p,s$, while for general quasicircles a Plemelj–Calderón decomposition holds under a Minkowski-content constraint $h(Γ)$. It combines interpolation, Plemelj theory, and transmission operators to unify boundary behavior across interior and exterior domains, revealing deep links between geometric regularity, weighted Sobolev traces, and Dirichlet-type energy principles in two dimensions.
Abstract
Let $Γ$ be a bounded Jordan curve and $Ω_i,Ω_e$ 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 $Ω_i$ 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.