Summary: The operator-valued Fourier multiplier theorems in E-valued weighted Lebesgue and Besov spaces are studied. These results permit us to show embedding theorems in weighted Besov-Lions type spaces \(B^{l,s}_{p,q,\gamma} (\Omega; E_0, E)\), where \(E_0\), \(E\) are two Banach spaces and \(E_0 \subset E\). The most regular class of interpolation space \(E_\alpha\), between \(E_0\) and \(E\) are found such that the mixed differential operator \(D^\alpha\) is bounded from \(B^{l,s}_{p,q,\gamma} (\Omega; E_0, E)\) to \(B^{s}_{p,q,\gamma} (\Omega; E_\alpha)\) and Ehrling-Nirenberg-Gagliardo type sharp estimates are established. By using these results the \(B^{s}_{p,q,\gamma}\)-separability properties of degenerate differential operators are studied.