OPERATOR-VALUED Lq → Lp FOURIER MULTIPLIERS

Fourier multiplier theorems provides one of the most important tools in the study of partial differential equations and embedding theorems. They are very often used to establish maximal regularity of elliptic and parabolic differential operator equations. Operator–valued multiplier theorems in Banach–valued function spaces have been discussed extensively in [1, 2, 3, 5, 7, 8, 9, 10, 11, 12 ]. Boundary value problems (BVPs) for differential–operator equations (DOEs) in Banach–valued function spaces investigated in [1, 7 and reference therein]. The exposition of Lp-multipliers and some related references, can be found in [11] , [12] and [6, §2.2.1-§2.2.4]. In the second and third sections we shall study FMT. Our aim is to establish multiplier theorems from Lq to Lp for q ≤ p extending those in [8] and [10]. In [8] author generalized Miklin’s theorem to the vector-valued case i.e. it is shown that a bounded function ψ : R → C is Fourier multiplier in Lp(X) provided X is UMD space and