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$L^{p}$-BOUNDS FOR PSEUDO-DIFFERENTIAL OPERATORS ON COMPACT LIE GROUPS
Published online by Cambridge University Press: 03 April 2017
Abstract
Given a compact Lie group $G$, in this paper we establish
$L^{p}$-bounds for pseudo-differential operators in
$L^{p}(G)$. The criteria here are given in terms of the concept of matrix symbols defined on the noncommutative analogue of the phase space
$G\times \widehat{G}$, where
$\widehat{G}$ is the unitary dual of
$G$. We obtain two different types of
$L^{p}$ bounds: first for finite regularity symbols and second for smooth symbols. The conditions for smooth symbols are formulated using
$\mathscr{S}_{\unicode[STIX]{x1D70C},\unicode[STIX]{x1D6FF}}^{m}(G)$ classes which are a suitable extension of the well-known
$(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D6FF})$ ones on the Euclidean space. The results herein extend classical
$L^{p}$ bounds established by C. Fefferman on
$\mathbb{R}^{n}$. While Fefferman’s results have immediate consequences on general manifolds for
$\unicode[STIX]{x1D70C}>\max \{\unicode[STIX]{x1D6FF},1-\unicode[STIX]{x1D6FF}\}$, our results do not require the condition
$\unicode[STIX]{x1D70C}>1-\unicode[STIX]{x1D6FF}$. Moreover, one of our results also does not require
$\unicode[STIX]{x1D70C}>\unicode[STIX]{x1D6FF}$. Examples are given for the case of
$\text{SU}(2)\cong \mathbb{S}^{3}$ and vector fields/sub-Laplacian operators when operators in the classes
$\mathscr{S}_{0,0}^{m}$ and
$\mathscr{S}_{\frac{1}{2},0}^{m}$ naturally appear, and where conditions
$\unicode[STIX]{x1D70C}>\unicode[STIX]{x1D6FF}$ and
$\unicode[STIX]{x1D70C}>1-\unicode[STIX]{x1D6FF}$ fail, respectively.
MSC classification
- Type
- Research Article
- Information
- Creative Commons
- This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
- Copyright
- © Cambridge University Press 2017
Footnotes
The first author was supported by the Leverhulme Research Grant RPG-2014-02. The second author was supported by the EPSRC Grant EP/K039407/1. No new data was collected or generated during the course of research.
References
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