Convolution of three functions by means of bilinear maps and applications

Let X, Y and E be complex Banach spaces, and let u: X x Y --> E be a bounde d bilinear map. If f, g are analytic functions in the unit disc taking valu es in X and Y with Taylor coefficients x(n) and y(n) respectively, we defin e the E-valued function f *(u) g whose Taylor coefficients are given by u(x (n), y(n)). Given two bounded bilinear maps, u: X x Y --> E and v: Z x E -- > F, in our main theorem we prove that Young's Theorem can be improved by s howing that the function f *(v) (g *(u) h) is in the Hardy space H-p(F) pro vided that f, g and h are in the Vector valued Besov spaces corresponding t o those that appear in some classical inequalities by Hardy-Littlewood and Littlewood-Paley. We also investigate the class of Banach spaces for which these inequalities hold in the vector setting, and we give a number of applications of our th eorem for these spaces and for certain bilinear maps (such as convolution, tensor products,...), obtaining results both in the scalar and the vector v alued cases.