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 $G$, that 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 $MA(\Gamma)\rightarrow MA(G)$ is not always continuous. We also show how Liao's Property $(T_{Schur}, G, K)$ opposes tame cuts. Some examples are provided.