On the growth of Fourier multipliers

Abstract

We define a sequence of functions, namely, tame cuts, in the Fourier algebra $A(G)$ of a locally compact group $G$, which satisfies certain convergence and growth conditions. This new consideration allows us to give a group admitting a Fourier multiplier that is not completely bounded. Furthermore, we show that the induction map $MA(\Gamma )\to MA(G)$ is not always continuous. We also show how Liao’s property $(T_{\mathrm{Schur}},G,K)$ opposes tame cuts. Some examples are provided.