In this paper, we shall introduce the concept of the Bessel (Riesz) potential Köthe function spacesXs (Xs) and give some dual estimates for a class of operators determined by a semi-group in the spacesLq (−T, T; Xs) (Lq(−T, T; Xs)). Moreover, some time-spaceLp−Lp′ estimates for the semi-group exp(it(-Δ)m/2) and the operatorA:=∫
exp(i(t-τ)(-Δ)m/2)·dτ in the Lebesgue-Besov spacesLq(−T,T;Bp,2s
are given. On the basis of these results, in a subsequent paper we shall present some further applications to a class of nonlinear wave equations.