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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


On a maximal function on compact Lie groups

Authors: Michael Cowling and Christopher Meaney
Journal: Trans. Amer. Math. Soc. 315 (1989), 811-822
MSC: Primary 43A75; Secondary 22E30, 42B25
MathSciNet review: 958889
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Abstract: Suppose that $ G$ is a compact Lie group with finite centre. For each positive number $ s$ we consider the $ \operatorname{Ad}(G)$-invariant probability measure $ {\mu _s}$ carried on the conjugacy class of $ \exp (s{H_\rho })$ in $ G$. This one-parameter family of measures is used to define a maximal function $ \mathcal{M}\,f$, for each continuous function $ f$ on $ G$. Our theorem states that there is an index $ {p_0}$ in $ (1,2)$, depending on $ G$, such that the maximal operator $ \mathcal{M}$ is bounded on $ {L^p}(G)$ when $ p$ is greater than $ {p_0}$. When the rank of $ G$ is greater than one, this provides an example of a controllable maximal operator coming from averages over a family of submanifolds, each of codimension greater than one.

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PII: S 0002-9947(1989)0958889-3
Article copyright: © Copyright 1989 American Mathematical Society