In this article, we investigate the relationships among Ahlfors n-regular domains, \({\dot{V}}^{s,\phi }\) -extension domains, and \({\dot{V}}^{s,\phi }\) -embedding domains. Specifically, we prove that if \(\Omega \) is an Ahlfors n-regular domain, then it is also a \({\dot{V}}^{s,\phi }\) -extension domain. Furthermore, assuming that \(\phi \) is doubling with \(K_\phi \in [1, 2^{\frac{n}{s}}) \cup (2^{\frac{n}{s}}, \infty ) \) , we establish that every \({\dot{V}}^{s,\phi }\) -extension domain is a \({\dot{V}}^{s,\phi }\) -embedding domain. Additionally, under the doubling condition, we demonstrate that every \({\dot{V}}^{s,\phi }\) -embedding domain is an Ahlfors n-regular domain. Consequently, under the doubling property with \(K_\phi \in [1,2^{\frac{n}{s}}) \cup (2^{\frac{n}{s}},\infty )\) , we conclude that the classes of Ahlfors n-regular domains, \({\dot{V}}^{s,\phi }\) -extension domains, and \({\dot{V}}^{s,\phi }\) -embedding domains are equivalent.
