Embeddings and Separable Differential Operators in Spaces of Sobolev–Lions type

We study embedding theorems for anisotropic spaces of Bessel–Lions type $H_{p,\gamma}^l(\Omega;E_0,E)$, where $E_0$ and $E$ are Banach spaces. We obtain the most regular spaces $E_\alpha$ for which mixed differentiation operators $D^\alpha$ from $H_{p,\gamma}^l(\Omega;E_0,E)$ to $L_{p,\gamma}(\Omega;E_\alpha)$ are bounded. The spaces $E_\alpha$ are interpolation spaces between $E_0$ and $E$, depending on $\alpha=(\alpha_1,\alpha_2,\dots,\alpha_n)$ and $l=(l_1,l_2,\dots,l_n)$. The results obtained are applied to prove the separability of anisotropic differential operator equations with variable coefficients.