On the growth of Fourier multipliers

TL;DR

This paper introduces tame cuts in Fourier algebras to analyze Fourier multipliers, revealing new insights into their boundedness and continuity properties on locally compact groups.

Contribution

It defines tame cuts in Fourier algebras and demonstrates their implications for Fourier multipliers and the induction map, providing new examples and counterexamples.

Findings

Existence of groups with Fourier multipliers that are not completely bounded.

Discontinuity of the induction map $MA(\Gamma) ightarrow MA(G)$ in some cases.

Liao's Property $(T_{Schur}, G, K)$ opposes tame cuts.

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.