Sidonicity in compact, abelian hypergroups

Kathryn Hare

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In this paper we study generalized Besov type spaces on the Laguerre hypergroup and we give some characterizations using different equivalent norms which allows to reach results of completeness, continuous embeddings and density of some subspaces. A generalized Calderón-Zygmund formula adapted to the harmonic analysis on the Laguerre Hypergroup is obtained inducing two more equivalent norms.

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Let $C V_{p} (F)$ be the left convolution operators on $L^{p} (G)$ with support included in F and $M_{p} (F)$ denote those which are norm limits of convolution by bounded measures in M(F). Conditions on F are given which insure that $C V_{p} (F)$ , $C V_{p} (F) / M_{p} (F)$ and $C V_{p} (F) / W$ are as big as they can be, namely have $l^{\infty}$ as a quotient, where the ergodic space W contains, and at times is very big relative to $M_{p} (F)$ . Other subspaces of $C V_{p} (F)$ are considered. These improve results of Cowling and Fournier, Price and Edwards, Lust-Piquard, and others.