Convolution of three functions by means of bilinear maps and applications

Abstract

Let $X$, $Y$ and $E$ be complex Banach spaces, and let $u:X \times Y \rightarrow E$ be a bounded bilinear map. If $f$, $g$ are analytic functions in the unit disc taking values in $X$ and $Y$ with Taylor coefficients $x_{n}$ and $y_{n}$ respectively, we define the $E$-valued function $f \ast_{u}\, g$ whose Taylor coefficients are given by $u(x_{n},y_{n})$. Given two bounded bilinear maps, $u:X \times Y \rightarrow E$ and $v:Z \times E \rightarrow F$, in our main theorem we prove that Young's Theorem can be improved by showing that the function $f \ast_{v}(g \ast_{u} h)$ is in the Hardy space $H^{p}(F)$ provided that $f$, $g$ and $h$ are in the vector valued Besov spaces corresponding to 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 theorem for these spaces and for certain bilinear maps (such as convolution, tensor products, $\ldots$) obtaining results both in the scalar and the vector valued cases.