Tame ${\displaystyle L^p}$-multipliers

We call an ${\displaystyle L^p}$-multiplier m tame if for each complex homomorphism χ acting on the space of ${\displaystyle L^p}$ multipliers there is some ${\displaystyle {\gamma }_0\in \Gamma }$ and |a| ≤ 1 such that ${\displaystyle \chi (\gamma m)=am({\gamma }_0\gamma )}$ for all γ ∈ Γ. Examples of tame multipliers include tame measures and one-sided Riesz products. Tame multipliers show an interesting similarity to measures. Indeed we show that the only tame idempotent multipliers are measures. We obtain quantitative estimates on the size of ${\displaystyle L^p}$-improving tame multipliers which are similar to those obtained for measures, but are false for non-tame multipliers. One-sided Riesz products are seen to play a similar role in the study of tame multipliers as Riesz products do in the study of measures.