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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Multiplier transformations on compact Lie groups and algebras


Author: Robert S. Strichartz
Journal: Trans. Amer. Math. Soc. 193 (1974), 99-110
MSC: Primary 22E30; Secondary 43A75
DOI: https://doi.org/10.1090/S0002-9947-1974-0357688-2
MathSciNet review: 0357688
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Abstract: Let G be a semisimple compact Lie group and $Tf = \sum \phi (m){d_{m \chi m}} \ast f$ a bi-invariant operator on ${L^2}(G)$, where ${\chi _m}$ and ${d_m}$ are the characters and dimensions of the irreducible representations of G, which are indexed by a lattice of points m in the Lie algebra $\mathfrak {G}$ in a natural way. If $\Phi$ is a bounded ad-invariant function on $\mathfrak {G}$ and \begin{equation}\tag {$\ast $} \phi {\text {(}}m{\text {) = }}\Phi {\text {(}}m{\text { + }}\beta {\text {)}}\quad {\text {or}}\end{equation} \begin{equation}\tag {$\ast \ast $} \phi {\text {(}}m{\text {) = }}\int _G {\Phi (m + \beta - {\text {ad}}\;g\beta )dg} \end{equation} $\beta$ being half the sum of the positive roots, then various properties of T are related to properties of the Fourier multiplier transformation on $\mathfrak {G}$ with multiplier $\Phi$. These properties include boundedness on ${L^1}$, uniform boundedness on ${L^p}$ of a family of operators, and, in the special case $G = {\text {SO}}(3)$, boundedness in ${L^p}$ for ad-invariant functions with $1 \leq p < 3/2$.


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Keywords: Multiplier transformation, compact Lie group, bi-invariant operator, ad-invariant function on the Lie algebra
Article copyright: © Copyright 1974 American Mathematical Society