Abstract capacity of regions and compact embedding with applications

The weighted Sobolev-Lions type spaces Wlp,γ(Ω;E0,E) =Wlp,γ(Ω;E)∩ Lp,γ(Ω;E0) are studied, where E0, E are two Banach spaces and E0 is continuously and densely embedded on E. A new concept of capacity of region Ω ∈ Rn in Wlp,γ(Ω;E0,E) is introduced. Several conditions in terms of capacity of region Ω and interpolations of E0 and E are found such that ensure the continuity and compactness of embedding operators. In particular, the most regular class of interpolation spaces Eα between E0 and E, depending of α and l, are found such that mixed differential operators Dα are bounded and compact from Wlp,γ(Ω;E0,E) to Eα-valued Lp,γ spaces. In applications, the maximal regularity for differential-operator equations with parameters are studied.