Operator-valued Fourier multipliers in vector-valued function spaces and application

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 Bl,s p,q,γ (Ω; E0, E), where E0, E are two Banach spaces and E0 ⊂ E. The most regular class of interpolation space Eα, between E0 and E are found such that the mixed differential operator Dα is bounded from Bl,s p,q,γ (Ω; E0, E) to Bs p,q,γ (Ω; Eα) and Ehrling-Nirenberg-Gagliardo type sharp estimates are established. By using these results the Bs p,q,γ−separability properties of degenerate differential operators are studied.