The weighted Sobolev-Lions type spaces W l p , γ (Ω;E 0 ,E) =W l p , γ (Ω;E)∩ L p , γ (Ω;E 0 ) are studied, where E 0 , E are two Banach spaces and E 0 is continuously and densely embedded on E. A new concept of capacity of region Ω ∈ R n in W l p , γ (Ω;E 0 ,E) is introduced. Several conditions in terms of capacity of region Ω and interpolations of E 0 and E are found such that ensure the continuity and compactness of embedding operators. In particular, the most regular class of interpolation spaces E α between E 0 and E, depending of α and l, are found such that mixed differential operators D α are bounded and compact from W l p , γ (Ω;E 0 ,E) to E α -valued L p,γ spaces. In applications, the maximal regularity for differential-operator equations with parameters are studied.