A remark on the asymmetry of convolution operators

Giulini, Saverio

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 83 (1989) no. 1, pp. 85-88

Résumé

A convolution operator, bounded on $L^{q}(\mathbb{R}^{n})$, is bounded on $L^{p}(\mathbb{R}^{n})$, with the same operator norm, if $p$ and $q$ are conjugate exponents. It is well known that this fact is false if we replace $\mathbb{R}^{n}$ with a general non-commutative locally compact group $G$. In this paper we give a simple construction of a convolution operator on a suitable compact group $G$, wich is bounded on $L^{q}(G)$ for every $q \in [2,\infty)$ and is unbounded on $L^{p}(G)$ if $p \in [1,2)$.