J. García-Cuerva and J. Parcet

The main purpose of this paper is to study the validity of the Hausdorff--Young inequality for vector-valued functions defined on a non-commutative compact group. As we explain in the introduction, the natural context for this research is that of operator spaces. This leads us to formulate a whole new theory of Fourier type and cotype for the category of operator spaces. The present paper is the first step in this program, where the basic theory is presented, the main examples are analyzed and some important questions are posed.

2000 Mathematical Subject Classification: 43A77, 46L07.

Keywords: operator space, Hausdorff--Young inequality, compact group, Fourier type and cotype.

E-mail:
jose.garcia-cuerva@uam.es
javier.parcet@uam.es