Fourier multipliers on extensions of Heisenberg type groups and the contraction from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> to <mml:math xmlns:mml="
Abstract
<jats:p> If <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mi>K</mml:mi> <mml:mi>A</mml:mi> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> is the Iwasawa decomposition of a rank one semisimple Lie group, then the semi-direct product <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> is a contraction of the compact group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> . We use this fact to transfer harmonic analysis on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> to harmonic analysis on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> , specifically transferring estimates for Fourier multipliers of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> . The critical novelty is a relationship between the representation theory of the two groups: we show how to embed the irreducible representations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> into the Fock spaces for the irreducible representations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> , and prove a limiting formula for the associated matrix entries. As an application we prove a version of de Leeuw’s theorem on transferring <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -multiplier estimates from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> . </jats:p>