Bessel (Riesz) potentials on Banach function spaces and their applications - I - Theory

In this paper, we shall introduce the concept of the Bessel (Riesz) potential Kothe function spaces X-s (X-s) and give some dual estimates for a class of operators determined by a semi-group in the spaces L-q(-T, T; X-s) (L-q(-T, T; (X) over dot (s))). Moreover, some time-space L-p - L-p' estimates for the semi-group exp(it(-Delta)(m/2)) and the operator A := integral(0)(t) exp(i(t - tau)(-Delta)(m/2)).d tau in the Lebesgue-Besov spaces L-q(-T, T; (B) over dot (s)(p,2)) 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.