The author introduces the function spaces \(D_\ell(\mathbb{R}^d)\) which are based on a regularity property for the critical Sobolev spaces \(W^{s,p}(\mathbb{R}^d)\) with \(sp=d\). The spaces \(D_\ell(\mathbb{R}^d)\) contain all the critical Sobolev spaces. The spaces \(D_\ell(\mathbb{R}^d)\) contain neither \(\text{BMO}(\mathbb{R}^d)\) nor \(\text{VMO}(\mathbb{R}^d)\). The main result of the author is that \(D_\ell(\mathbb{R}^d)\) is embedded in \(\text{BMO}(\mathbb{R}^d)\). To this end, the author shows that \(D_\ell(\mathbb{R}^d)\) is embedded in \(D_1(\mathbb{R}^d)\) and that \(D_1(\mathbb{R}^d)\) is embedded in \(\text{BMO}(\mathbb{R}^d)\). The paper ends with considerations about further problems in the study of the spaces \(D_\ell(\mathbb{R}^d)\).