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.
Citation
José Luis Arregui. Oscar Blasco. "Convolution of three functions by means of bilinear maps and applications." Illinois J. Math. 43 (2) 264 - 280, Summer 1999. https://doi.org/10.1215/ijm/1255985214
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