Fourier multipliers on extensions of Heisenberg type groups and the contraction from K to N ¯ M

This is the authors' abstract. We don't add key points for this paper.

If G = K A N is the Iwasawa decomposition of a rank one semisimple Lie group, then the semi-direct product M N is a contraction of the compact group K . We use this fact to transfer harmonic analysis on K to harmonic analysis on M N , 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 M N , 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 M N .