Suppose \(Q\subset {\mathbb {R}}^{d}\) is a cube and \( W^{s_{0},p_{0}}\) , \(W^{\sigma ,q}\) , \(W^{s_{1},p_{1}}\) are three Sobolev-Slobodeckiĭ spaces such that \(s_{0}, s_{1}\ge 0\) , \(p_{0}, p_{1}\in [1,\infty ]\) and \(\sigma :=\left( 1-\theta \right) s_0+\theta s_1\) , \(1/q:=\) \( (1-\theta )/p_0+\theta /p_1\) for some \(\theta \in (0,1)\) . We give a necessary and sufficient condition for the embedding * \(\begin{aligned} W^{\sigma ,q}(Q) \hookrightarrow W^{s_{0},p_{0}}(Q)+W^{s_{1},p_{1}}(Q), \end{aligned}\) to hold. By this we complement some results of Mironescu (Sum-Intersection Property of Sobolev Spaces. vol 135. Springer, Cham. [12]). We also show the connection of our results with the Gagliardo–Nirenberg noninequalities proved by Brezis and Mironescu (Gagliardo-Nirenberg inequalities and non-inequalities: the full story. 35(5): 355–1376 [4]) and, in the case ( \( *\) ) does not hold, we indicate the construction of some counterexamples.
