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$L^p$-bounds for pseudo-differential operators on compact Lie groups

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Title: $L^p$-bounds for pseudo-differential operators on compact Lie groups
Authors: Delgado Valencia, JC
Ruzhansky, M
Item Type: Journal Article
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.
Issue Date: May-2019
Date of Acceptance: 25-Jan-2017
URI: http://hdl.handle.net/10044/1/43750
DOI: https://doi.org/10.1017/S1474748017000123
ISSN: 1474-7480
Publisher: Cambridge University Press (CUP)
Start Page: 531
End Page: 559
Journal / Book Title: Journal of the Institute of Mathematics of Jussieu
Volume: 18
Issue: 3
Copyright Statement: © Cambridge University Press 2017. 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. doi:10.1017/S1474748017000123
Sponsor/Funder: The Leverhulme Trust
Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: RPG-2014-002
Keywords: Science & Technology
Physical Sciences
compact Lie groups
pseudo-differential operators
L-p bounds
Primary 35S05, Secondary 22E30, 47G30
General Mathematics
Publication Status: Published
Online Publication Date: 2017-04-03
Appears in Collections:Pure Mathematics
Faculty of Natural Sciences