If $G=KAN$ is the Iwasawa decomposition of a rank one semisimple Lie group, then the semi-direct product $MN$ is a contraction of the compact group $K$. We use this fact to transfer harmonic analysis on $K$ to harmonic analysis on $MN$, specifically transferring estimates for Fourier multipliers of $L^p$. The critical novelty is a relationship between the representation theory of the two groups: we show how to embed the irreducible representations of $K$ into the Fock spaces for the irreducible representations of $MN$, and prove a limiting formula for the associated matrix entries. As an application we prove a version of de Leeuw’s theorem on transferring $L^p$-multiplier estimates from $K$ to $MN$.
Accepted:
Published online:
Dooley, Anthony H. 1 ; Gupta, Sanjiv Kumar 2
Dooley, Anthony H.; Gupta, Sanjiv Kumar. Fourier multipliers on extensions of Heisenberg type groups and the contraction from $K$ to $\bar{N}M$. Journal of Lie Theory, Volume 36 (2026) no. 2-3, pp. 561-584. doi: 10.5802/jolt.1440
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author = {Dooley, Anthony H. and Gupta, Sanjiv Kumar},
title = {Fourier multipliers on extensions of {Heisenberg} type groups and the contraction from $K$ to $\bar{N}M$},
journal = {Journal of Lie Theory},
pages = {561--584},
year = {2026},
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