On the interpolation space $(L^p(\mathrm{\Omega }),W^{1,p}(\mathrm{\Omega }){)}_{s,p}$ in non-smooth domains

We show that, for certain non-smooth bounded domains $\mathrm{\Omega }\subset {\mathbb{R}}^n$, the real interpolation space $(L^p(\Omega), W^{1,p}(\Omega)){s,p}$isthesubspace$\widetilde W^{s,p}(\Omega) \subset L^p(\Omega)$inducedbytherestrictedfractionalseminorm$$ |f|{\widetilde W^{s,p}(\Omega)} = \Big( \int_\Omega \int_{|x-y|<\frac{d(x)}2} \frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}} , dy ,dx \Big)^\frac{1}{p}. $$ In particular, the above result includes simply connected uniform domains in the plane, for which a characterization of the interpolation space was previously unknown.