Separable anisotropic differential operators in weighted abstract spaces and applications ✩

In this paper operator-valued multiplier theorems in Banach-valued weighted Lp spaces are studied. Also weighted Sobolev– Lions type spaces Wl p,γ (Ω;E0,E) = Wl p,γ (Ω;E) ∩ Lp,γ (Ω;E0) are discussed when E0, E are two Banach spaces and E0 is continuously and densely embedded on E. Several conditions are found that ensure the continuity of the embedding operators that are optimally regular in these spaces in terms of interpolations of E0. These results permit us to show the separability of the anisotropic differential operators in an E-valued weighted Lp space. By using these results the maximal regularity of infinite systems of quasi elliptic partial differential equations are established. © 2007 Elsevier Inc. All rights reserved.