Fractional Besov-Sobolev Spaces on Chord-arc Curves

Abstract

Let $\Gamma$ be a bounded rectifiable Jordan curve, with complementary domains ${\Omega }_i$ and ${\Omega }_e$. For $1<p<\infty$ and $0<s<1$, we study the spaces ${\mathcal{B}}_{p,p}^s(\Omega )$ of harmonic functions satisfying \[ \iint_Ω|\nabla u(z)|^p \operatorname{dist}(z,Γ)^{(1-s)p-1}\,dA(z)<\infty, \qquad Ω\in\{Ω_i,Ω_e\}. \] When $\Gamma$ is the unit circle, the traces of these two spaces coincide with the fractional Besov-Sobolev space $B_{p,p}^s(\Gamma )$. The equivalence of boundary seminorms and harmonic energies on the circle raises a geometric question: which rough curves preserve these estimates? For every radial-Lipschitz curve $\Gamma$, we prove \[ \operatorname{Tr}_Γ\mathcal B_{p,p}^s(Ω_i) =B_{p,p}^s(Γ) =\operatorname{Tr}_Γ\mathcal B_{p,p}^s(Ω_e) \] whenever either $1<p<2$ and $2/p-1\le s<1$, $2<p<\infty$ and $0<s\le 1/2+1/p$ or $p=2$ and $0<s<1$. For arbitrary chord-arc curves, the same trace identities hold when $1/p\le s<1/2+1/(2p)$. Taking $s=1-1/p$ yields an almost-Dirichlet principle for the unweighted $p$-energy on radial-Lipschitz domains when $3/2\le p\le 4$. Here, the upper bound $4$ is sharp. In particular, every Riemann map $g:{\Omega }_i\to {\mathbb{D}}_i$ from a bounded radial-Lipschitz domain satisfies $g^{\prime }\in L^4({\Omega }_i)$, and the exponent $4$ is optimal uniformly over this class. This gives a sharp endpoint counterpart of Brennan's integrability conclusion for radial-Lipschitz domains.

arXiv comment56 pages, 3 figures