On a translation property of positive definite functions
Volume 89 / 2010
Abstract
If $G$ is a locally compact group with a compact invariant neighbourhood of the identity $e$, the following property $(*)$ holds: For every continuous positive definite function $h\geq 0$ with compact support there is a constant $C_h>0$ such that $\int L_xh\cdot g\le C_h\int hg$ for every continuous positive definite $g\ge 0$, where $L_x$ is left translation by $x$. In [L], property $(*)$ was stated, but the above inequality was proved for special $h$ only. That “for one $h$” implies “for all $h$” seemed obvious, but turned out not to be obvious at all. We fill this gap by means of a new structure theorem for IN-groups. For $p\in {\mathbb N}$ even, property $(*)$ easily implies the following property $(*)_p$: For every relatively compact invariant neighbourhood $U$ of $e$, there is a constant $C_U>0$ such that ${\|\chi_{xU}\cdot g\|}_p\le C_U{\|\chi_U\cdot g\|}_p$ for every continuous positive definite function $g$. For all other $p\in (1,\infty )$, property $(*)_p$ fails (see [L]). In the special case of the unit circle, the ${\|\ \|}_p$-norm results are essentially due to N. Wiener, S. Wainger, and H. S. Shapiro. For compact abelian groups they are due to M. Rains, and for locally compact abelian groups to J. Fournier.
