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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
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

$\displaystyle \phi {\text{(}}m{\text{) = }}\Phi {\text{(}}m{\text{ + }}\beta {\text{)}}\quad{\text{or}}$ ($ \ast$)

$\displaystyle \phi {\text{(}}m{\text{) = }}\int_G {\Phi (m + \beta - {\text{ad}}\;g\beta )dg}$ ($ \ast \ast$)

$ \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