Anisotropic Fractional Sobolev Space Restricted on a Bounded Domain

This paper studies the anisotropic fractional Sobolev space restricted on a bounded domain in the Euclidean space \(\mathbb {R}^{n}\) ${\mathbb{R}}^n$ with fractional order \(\alpha \in [0,n]\) $\alpha \in [0,n]$ , which complements the previous theory with fractional order \(\alpha >n\) $\alpha >n$ . We investigate the seminorm of the characteristic function as the anisotropic fractional perimeter restricted on a bounded domain, and systematically establish its metric properties including the upper bound estimation. For application, we prove the embedding law with respect to the anisotropic fractional Sobolev space and the Radon measure based Lebesgue space restricted on a bounded domain by the intrinsic geometric characterization.