Espaces de Sobolev fractionnaires sur les quasicercles

par Michel Zinsmeister

mardi 10 févr. 2026, 14:00 → 15:00 Europe/Paris

Salle de séminaire (Orléans institut Denis Poisson)

Let $\Gamma$ be a bounded Jordan curve and $\Omega_{i},\Omega_{e}$ its two complementary components. For $1<p<\infty$, $s\in(0,1)$ we define $\mathcal{B}^{s}_{p,p}(\Omega_{i,e})$ as the set of functions $f:\Gamma\to\mathbb{C}$ having harmonic extension $u$ respectively in $\Omega_{i}$ and $\Omega_{e}$ such that

$$
\iint_{\Omega_{i,e}}|\nabla u(z)|^{p}d(z,\Gamma)^{(1-s)p-1}dxdy<+\infty.
$$

If $\Gamma$ is further assumed to be rectifiable we define $B^{s}_{p,p}(\Gamma)$ as the space of functions $f\in L^{p}(\Gamma)$ such that

$$
\iint_{\Gamma\times\Gamma}\frac{|f(z)-f(\zeta)|^{p}}{|z-\zeta|^{1+ps}}|dz||d\zeta|<+\infty.
$$

When $\Gamma$ 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 $s=1/p$, $p\geq 2$, 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 $\mathcal{B}^{s}_{p,p}(\Omega_{i,e})$ and give conditions for equality of these trace-spaces that we then call $\mathcal{B}^{s}_{p,p}(\Gamma)$. Using Plemelj-Calderón property we further identify $\mathcal{B}^{s}_{p,p}(\Gamma)$ 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 Mazya.