Separable anisotropic differential operators in weighted abstract spaces and applications

In this paper operator-valued multiplier theorems in Banach-valued weighted L-p spaces are studied. Also weighted Sobolev-Lions type spaces W-p,gamma(l)(Omega; E-0, E) = W-p,gamma(l)(Omega; E) boolean AND L-p,L-gamma (Omega; E-0) are discussed when E-0, E are two Banach spaces and E-0 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 E-0. 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. (C) 2007 Elsevier Inc. All rights reserved.