Vector Valued Fourier Transforms and Fourier Type
The central question of this talk is whether the methods of
scalar-valued harmonic analysis can be transferred to the
vector-valued setting. Fortunately, the answer of this question is
negative. This opens the opportunity to classify Banach spaces (or
operators) by measuring how well scalar-valued problems can be
extended. More precisely, we ask if for a given Banach space $X$
and a locally compact Abelian group $G$, the $X$-valued Fourier
transform on $G$ still satisfies a Hausdorff-Young inequality.
Banach spaces having this property are said to be of Fourier type
$p$ with respect to $G$. We will outline the theory around this
property and describe new results in this direction.
Mariusz Piotrowski (CMUC/U. Coimbra)
November 18, 2005