On the Hausdorff–Young theorem for commutative hypergroups
Volume 131 / 2013
Colloquium Mathematicum 131 (2013), 219-231
MSC: Primary 43A32, 43A62; Secondary 43A15.
DOI: 10.4064/cm131-2-5
Abstract
We study the Hausdorff–Young transform for a commutative hypergroup $K$ and its dual space $\hat{K}$ by extending the domain of the Fourier transform so as to encompass all functions in $L^p(K,m)$ and $L^p(\hat{K},\pi)$ respectively, where $1\leq p \leq 2$. Our main theorem is that those extended transforms are inverse to each other. In contrast to the group case, this is not obvious, since the dual space $\hat{K}$ is in general not a hypergroup itself.
