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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 EP/K039407/1 |
Keywords: | Science & Technology Physical Sciences Mathematics compact Lie groups pseudo-differential operators L-p bounds MULTIPLIERS INEQUALITY SPACES math.AP math.AP Primary 35S05, Secondary 22E30, 47G30 General Mathematics |
Publication Status: | Published |
Online Publication Date: | 2017-04-03 |
Appears in Collections: | Pure Mathematics Mathematics Faculty of Natural Sciences |