Operator-valued Fourier Multipliers in Besov Spaces and Its Applications

In recent years, Fourier multiplier theorems in vector–valued function spaces have found many applications in embedding theorems of abstract function spaces and in theory of differential operator equations, especially in maximal regularity of parabolic and elliptic differential–operator equations. Operator–valued multiplier theorems in Banach–valued function spaces have been discussed extensively in [3,8,12,15, 17]. Boundary value problems (BVPs) for differential–operator equations (DOEs) in H–valued (Hilbert valued) function spaces and parabolic type convolution operator equations (COEs) with bounded operator coefficient have been studied in [1,2,6,7,9,13,14], and [3] respectively. Let E be a Banach space, x = (x1, x2, · · · , xn) ∈ Ω ⊂ R. Lr(Ω;E) denotes the space of all strongly measurable E–valued functions that are defined on the measurable subset Ω ⊂ R with the norm