Fractional Besov-Sobolev Spaces on Quasicircles

Abstract

Let $\Gamma$ be a bounded Jordan curve and $\Omega_i,\Omega_e$ its two complementary components. For $p\in (1, \infty),\,s\in(0,1)$ we define the two spaces $\mathcal{B}_{p,p}^s(\Omega_{i,e})$ as the set of harmonic functions $u$ respectively in $\Omega_i$ and $\Omega_e$ such that $$ \iint_{\Omega_{i,e}} |\nabla u(z)|^p d(z,\Gamma)^{(1-s)p-1} dxdy<+\infty.$$ When it is possible to identify these spaces with spaces of functions on the boundary (trace spaces), we address the question of their equality. When $\Gamma$ is the unit circle, these two spaces coincide with homogeneous fractional Besov-Sobolev spaces and the framework of quasicircles appears to be an appropriate generalization. In this framework, we study the boundedness of the Plemelj-Calder\'on operator and apply the results to show that for some values of $p,s$, if the two spaces coincide, they are restrictions to $\Gamma$ of some weighted Sobolev space. If $\Gamma$ is further assumed to be rectifiable, we define $B_{p,p}^s(\Gamma)$ as the space of functions $f\in L^p(\Gamma)$ such that $$\iint_{\Gamma\times \Gamma}\frac{|f(z)-f(\zeta)|^p}{|z-\zeta|^{1+ps}} |dz||d\zeta|<+\infty.$$ Again, these spaces coincide with the homogeneous fractional Besov-Sobolev spaces for the unit circle. While the chord-arc property is the necessary and sufficient condition for the equality $$\mathcal{B}_{p,p}^s(\Omega_{i})=\mathcal{B}_{p,p}^s(\Omega_{e})=B_{p,p}^s(\Gamma)$$ in the case of $s=1/p,\, p\ge 2$, this is no longer the case for general $s\in (0,1)$. However, we show that equality holds for radial-Lipschitz curves. Finally, we re-interpretate some of our results as some "almost"-Dirichlet principle in the spirit of Maz'ya.