Fractional Besov-Sobolev Spaces on Chord-arc Curves

Let $Γ$ be a bounded rectifiable Jordan curve, with complementary domains $Ω_i$ and $Ω_e$. For $1<p<\infty$ and $0<s<1$, we study the spaces $\mathcal B_{p,p}^s(Ω)$ 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_