Global quantization of pseudo-differential operators on compact Lie groups, SU(2), 3-sphere, and homogeneous spaces

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Global quantization of pseudo-differential operators on compact Lie groups, SU(2), 3-sphere, and homogeneous spaces. / Turunen, Ville; Ruzhansky, Michael.

In: INTERNATIONAL MATHEMATICS RESEARCH NOTICES, Vol. 2013, No. 11, 2013, p. 2439-2496.

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@article{60dba844f6e045cca8c8d3db851096ce,
title = "Global quantization of pseudo-differential operators on compact Lie groups, SU(2), 3-sphere, and homogeneous spaces",
abstract = "Global quantization of pseudo-differential operators on general compact Lie groups G is introduced relying on the representation theory of the group rather than on expressions in local coordinates. A new class of globally defined symbols is introduced and related to the usual H{\"o}rmander’s classes of operators Ψm(G). Properties of the new class and symbolic calculus are analyzed. Properties of symbols as well as L2-boundedness and Sobolev L2-boundedness of operators in this global quantization are established on general compact Lie groups. Operators on the three-dimensional sphere Graphic and on group SU(2) are analyzed in detail. An application is given to pseudo-differential operators on homogeneous spaces K\G. In particular, using the obtained global characterization of pseudo-differential operators on Lie groups, it is shown that every pseudo-differential operator in Ψm(K\G) can be lifted to a pseudo-differential operator in Ψm(G), extending the known results on invariant partial differential operators.",
author = "Ville Turunen and Michael Ruzhansky",
year = "2013",
doi = "10.1093/imrn/rns122",
language = "English",
volume = "2013",
pages = "2439--2496",
journal = "INTERNATIONAL MATHEMATICS RESEARCH NOTICES",
issn = "1073-7928",
number = "11",

}

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TY - JOUR

T1 - Global quantization of pseudo-differential operators on compact Lie groups, SU(2), 3-sphere, and homogeneous spaces

AU - Turunen, Ville

AU - Ruzhansky, Michael

PY - 2013

Y1 - 2013

N2 - Global quantization of pseudo-differential operators on general compact Lie groups G is introduced relying on the representation theory of the group rather than on expressions in local coordinates. A new class of globally defined symbols is introduced and related to the usual Hörmander’s classes of operators Ψm(G). Properties of the new class and symbolic calculus are analyzed. Properties of symbols as well as L2-boundedness and Sobolev L2-boundedness of operators in this global quantization are established on general compact Lie groups. Operators on the three-dimensional sphere Graphic and on group SU(2) are analyzed in detail. An application is given to pseudo-differential operators on homogeneous spaces K\G. In particular, using the obtained global characterization of pseudo-differential operators on Lie groups, it is shown that every pseudo-differential operator in Ψm(K\G) can be lifted to a pseudo-differential operator in Ψm(G), extending the known results on invariant partial differential operators.

AB - Global quantization of pseudo-differential operators on general compact Lie groups G is introduced relying on the representation theory of the group rather than on expressions in local coordinates. A new class of globally defined symbols is introduced and related to the usual Hörmander’s classes of operators Ψm(G). Properties of the new class and symbolic calculus are analyzed. Properties of symbols as well as L2-boundedness and Sobolev L2-boundedness of operators in this global quantization are established on general compact Lie groups. Operators on the three-dimensional sphere Graphic and on group SU(2) are analyzed in detail. An application is given to pseudo-differential operators on homogeneous spaces K\G. In particular, using the obtained global characterization of pseudo-differential operators on Lie groups, it is shown that every pseudo-differential operator in Ψm(K\G) can be lifted to a pseudo-differential operator in Ψm(G), extending the known results on invariant partial differential operators.

U2 - 10.1093/imrn/rns122

DO - 10.1093/imrn/rns122

M3 - Article

VL - 2013

SP - 2439

EP - 2496

JO - INTERNATIONAL MATHEMATICS RESEARCH NOTICES

JF - INTERNATIONAL MATHEMATICS RESEARCH NOTICES

SN - 1073-7928

IS - 11

ER -

ID: 9889523