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June  2020, 19(6): 3341-3366. doi: 10.3934/cpaa.2020148

On the Sobolev embedding properties for compact matrix quantum groups of Kac type

Department of Mathematics Education, Seoul National University, Gwanak-ro 1, Gwanak-gu, Seoul, 08826, Republic of Korea

Received  September 2019 Revised  December 2019 Published  March 2020

Fund Project: This research was supported by the Natural Sciences and Engineering Research Council of Canada and by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1C1C1A01009681)

We study the optimal order of natural analogues of Sobolev embedding properties within the framework of compact matrix quantum groups of Kac type. One of the main results of this paper is that the optimal order is given by the polynomial growth order of dual discrete quantum groups in a broad class, which covers all connected compact Lie groups, duals of polynomially growing discrete groups, $ O_2^+ $ and $ S_4^+ $. Outside the realm of co-amenable compact quantum groups, we prove that the optimal order is $ 3 $ for duals of free groups and free quantum groups $ O_N^+ $ and $ S_N^+ $, and that Sobolev embedding properties can be generalized for all compact matrix quantum groups of Kac type whose duals have the rapid decay property. In addition, we generalize sharpened Hausdorff-Young inequalities, compute degrees of the rapid decay property for duals of $ O_N^+,S_N^+ $ and prove sharpness of Hardy-Littlewood inequalities on duals of free groups.

Citation: Sang-Gyun Youn. On the Sobolev embedding properties for compact matrix quantum groups of Kac type. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3341-3366. doi: 10.3934/cpaa.2020148
References:
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R. AkylzhanovS. Majid and M. Ruzhansky, Smooth dense subalgebras and Fourier multipliers on compact quantum groups, Commun. Math. Phys., 362 (2018), 761-799.  doi: 10.1007/s00220-018-3219-4.  Google Scholar

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R. Kh. AkylzhanovE. D. Nursultanov and M. V. Ruzhansky, Hardy-littlewood-paley-type inequalities on compact Lie groups, Math. Notes, 100 (2016), 309-312.  doi: 10.4213/mzm11052.  Google Scholar

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R. AkylzhanovE. Nursultanov and M. Ruzhansky, Hardy-Littlewood-Paley inequalities and Fourier multipliers on $SU(2)$, Studia Math., 234 (2016), 1-29.  doi: 10.4064/sm8106-4-2016.  Google Scholar

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R. AkylzhanovM. Ruzhansky and E. Nursultanov, Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and $L^p$-$L^q$ Fourier multipliers on compact homogeneous manifolds, J. Math. Anal. Appl., 479 (2019), 1519-1548.  doi: 10.1016/j.jmaa.2019.07.010.  Google Scholar

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D. BakryT. CoulhonM. Ledoux and L. Saloff-Coste, Sobolev inequalities in disguise, Indiana Univ. Math. J., 44 (1995), 1033-1074.  doi: 10.1512/iumj.1995.44.2019.  Google Scholar

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T. Banica and R. Vergnioux, Growth estimates for discrete quantum groups, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 12 (2009), 321-340.  doi: 10.1142/S0219025709003677.  Google Scholar

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M. Brannan, Approximation properties for free orthogonal and free unitary quantum groups, J. Reine Angew. Math., 672 (2012), 223-251.  doi: 10.1515/crelle.2011.166.  Google Scholar

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M. Brannan and R. Vergnioux, Orthogonal free quantum group factors are strongly 1-bounded, Adv. Math., 329 (2018), 133-156.  doi: 10.1016/j.aim.2018.02.007.  Google Scholar

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E. Breuillard and E. Le Donne, On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry, Proc. Natl. Acad. Sci. USA, 110 (2013), 19220-19226.  doi: 10.1073/pnas.1203854109.  Google Scholar

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U. FranzG. HongF. LemeuxM. Ulrich and H. Zhang, Hypercontractivity of heat semigroups on free quantum groups, J. Oper. Theory, 77 (2017), 61-76.  doi: 10.7900/jot.2015nov13.2126.  Google Scholar

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A. Freslon and R. Vergnioux, The radial MASA in free orthogonal quantum groups, J. Funct. Anal., 271 (2016), 2776-2807.  doi: 10.1016/j.jfa.2016.08.007.  Google Scholar

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S. Giulini and G. Travaglini, $L^p$-estimates for matrix coefficients of irreducible representations of compact groups, Proc. Amer. Math. Soc., 80 (1980), 448-450.  doi: 10.2307/2043737.  Google Scholar

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A. González-PérezM. Junge and J. Parcet, Smooth Fourier multipliers in group algebras via Sobolev dimension, Ann. Sci. Ec. Norm. Super., 50 (2017), 879-925.  doi: 10.24033/asens.2334.  Google Scholar

[23]

U. Haagerup, An example of a nonnuclear $C^*$-algebra, which has the metric approximation property, Invent. Math., 50 (1978/79), 279-293.  doi: 10.1007/BF01410082.  Google Scholar

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Y. Isono, Examples of factors which have no Cartan subalgebras, Trans. Amer. Math. Soc., 367 (2015), 7917-7937.  doi: 10.1090/tran/6321.  Google Scholar

[25]

Y. Isono, Some prime factorization results for free quantum group factors, J. Reine Angew. Math., 722 (2017), 215-250.  doi: 10.1515/crelle-2014-0056.  Google Scholar

[26]

Y. Jiao and M. Wang, Noncommutative harmonic analysis on semigroups, Indiana Univ. Math. J., 66 (2017), 401-417.  doi: 10.1512/iumj.2017.66.6020.  Google Scholar

[27]

M. Junge and T. Mei, Noncommutative Riesz transforms-a probabilistic approach, Amer. J. Math., 132 (2010), 611-680.  doi: 10.1353/ajm.0.0122.  Google Scholar

[28]

M. Junge and T. Mei, BMO spaces associated with semigroups of operators, Math. Ann., 352 (2012), 691-743.  doi: 10.1007/s00208-011-0657-0.  Google Scholar

[29]

M. Junge, T. Mei and J. Parcet, An invitation to harmonic analysis associated with semigroups of operators, In Harmonic analysis and partial differential equations, Contemp. Math., Vol. 612, American Mathematical Society, Providence, RI, (2014), 107–122. Google Scholar

[30]

M. JungeT. Mei and J. Parcet, Smooth Fourier multipliers on group von Neumann algebras, Geom. Funct. Anal., 24 (2014), 1913-1980.  doi: 10.1007/s00039-014-0307-2.  Google Scholar

[31]

M. Junge, C. Palazuelos, J. Parcet and M. Perrin, Hypercontractivity in group von Neumann algebras, Mem. Amer. Math. Soc., 249 (2017), xii+83. doi: 10.1090/memo/1183.  Google Scholar

[32]

J. Kustermans and S. Vaes, Locally compact quantum groups, Ann. Sci. Ec. Norm. Super., 33 (2000), 837-934.  doi: 10.1016/S0012-9593(00)01055-7.  Google Scholar

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J. Kustermans and S. Vaes, Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand., 92 (2003), 68-92.  doi: 10.7146/math.scand.a-14394.  Google Scholar

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T. Mei and M. de la Salle, Complete boundedness of heat semigroups on the von Neumann algebra of hyperbolic groups, Trans. Amer. Math. Soc., 369 (2017), 5601-5622.  doi: 10.1090/tran/6825.  Google Scholar

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T. Mei and Éric Ricard, Free Hilbert transforms, Duke Math. J., 166 (2017), 2153-2182.  doi: 10.1215/00127094-2017-0007.  Google Scholar

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B. Nica, On the degree of rapid decay, Proc. Amer. Math. Soc., 138 (2010), 2341-2347.  doi: 10.1090/S0002-9939-10-10289-5.  Google Scholar

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B. Nica, On operator norms for hyperbolic groups, J. Topol. Anal., 9 (2017), 291-296.  doi: 10.1142/S179352531750008X.  Google Scholar

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G. Pisier and Q. Xu, Non-commutative $L^p$-spaces, In Handbook of the Geometry of Banach Spaces, Vol. 2, North-Holland, Amsterdam, (2003), 1459–1517. Google Scholar

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Éric Ricard and Q. Xu, Complex interpolation of weighted noncommutative $L_p$-spaces, Houston J. Math., 37 (2011), 1165-1179.   Google Scholar

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T. Timmermann, An invitation to Quantum Groups and Duality, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2008. From Hopf algebras to multiplicative unitaries and beyond. doi: 10.4171/043.  Google Scholar

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A. V. Daele and S. Wang, Universal quantum groups, Int. J. Math., 7 (1996), 255-263.  doi: 10.1142/S0129167X96000153.  Google Scholar

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N. Th. Varopoulos, Hardy-Littlewood theory for semigroups, J. Funct. Anal., 63 (1985), 240-260.  doi: 10.1016/0022-1236(85)90087-4.  Google Scholar

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N. Th. Varopoulos, Analysis on Lie groups, J. Funct. Anal., 76 (1988), 346-410.  doi: 10.1016/0022-1236(88)90041-9.  Google Scholar

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R. Vergnioux and C. Voigt, The $K$-theory of free quantum groups, Math. Ann., 357 (2013), 355-400.  doi: 10.1007/s00208-013-0902-9.  Google Scholar

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S. L. Woronowicz, Compact matrix pseudogroups, Commun. Math. Phys., 111 (1987), 613-665.   Google Scholar

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S. L. Woronowicz, Twisted $SU(2)$ group. An example of a noncommutative differential calculus, Publ. Res. Inst. Math. Sci., 23 (1987), 117-181.  doi: 10.2977/prims/1195176848.  Google Scholar

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show all references

References:
[1]

R. AkylzhanovS. Majid and M. Ruzhansky, Smooth dense subalgebras and Fourier multipliers on compact quantum groups, Commun. Math. Phys., 362 (2018), 761-799.  doi: 10.1007/s00220-018-3219-4.  Google Scholar

[2]

R. Kh. AkylzhanovE. D. Nursultanov and M. V. Ruzhansky, Hardy-littlewood-paley-type inequalities on compact Lie groups, Math. Notes, 100 (2016), 309-312.  doi: 10.4213/mzm11052.  Google Scholar

[3]

R. AkylzhanovE. Nursultanov and M. Ruzhansky, Hardy-Littlewood-Paley inequalities and Fourier multipliers on $SU(2)$, Studia Math., 234 (2016), 1-29.  doi: 10.4064/sm8106-4-2016.  Google Scholar

[4]

R. Akylzhanov and M. Ruzhansky, $L^p$-$L^q$ multipliers on locally compact groups, J. Funct. Anal., 278 (2020), 108324. doi: 10.1016/j.jfa.2019.108324.  Google Scholar

[5]

R. AkylzhanovM. Ruzhansky and E. Nursultanov, Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and $L^p$-$L^q$ Fourier multipliers on compact homogeneous manifolds, J. Math. Anal. Appl., 479 (2019), 1519-1548.  doi: 10.1016/j.jmaa.2019.07.010.  Google Scholar

[6]

D. Applebaum, Probability on Compact Lie Groups, Probability Theory and Stochastic Modelling, Vol. 70, Springer, Cham, 2014. With a foreword by Herbert Heyer. doi: 10.1007/978-3-319-07842-7.  Google Scholar

[7]

D. BakryT. CoulhonM. Ledoux and L. Saloff-Coste, Sobolev inequalities in disguise, Indiana Univ. Math. J., 44 (1995), 1033-1074.  doi: 10.1512/iumj.1995.44.2019.  Google Scholar

[8]

T. Banica and R. Vergnioux, Growth estimates for discrete quantum groups, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 12 (2009), 321-340.  doi: 10.1142/S0219025709003677.  Google Scholar

[9]

J. Bergh and J. Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[10]

M. Brannan, Approximation properties for free orthogonal and free unitary quantum groups, J. Reine Angew. Math., 672 (2012), 223-251.  doi: 10.1515/crelle.2011.166.  Google Scholar

[11]

M. BrannanB. Collins and R. Vergnioux, The Connes embedding property for quantum group von Neumann algebras, Trans. Amer. Math. Soc., 369 (2017), 3799-3819.  doi: 10.1090/tran/6752.  Google Scholar

[12]

M. Brannan and R. Vergnioux, Orthogonal free quantum group factors are strongly 1-bounded, Adv. Math., 329 (2018), 133-156.  doi: 10.1016/j.aim.2018.02.007.  Google Scholar

[13]

E. Breuillard and E. Le Donne, On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry, Proc. Natl. Acad. Sci. USA, 110 (2013), 19220-19226.  doi: 10.1073/pnas.1203854109.  Google Scholar

[14]

T. BrunoM. M. PelosoA. Tabacco and M. Vallarino, Sobolev spaces on Lie groups: embedding theorems and algebra properties, J. Funct. Anal., 276 (2019), 3014-3050.  doi: 10.1016/j.jfa.2018.11.014.  Google Scholar

[15]

M. CowlingS. Giulini and S. Meda, $L^p$-$L^q$ estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces I, Duke Math. J., 72 (1993), 109-150.  doi: 10.1215/S0012-7094-93-07206-7.  Google Scholar

[16]

P. de la Harpe, Groupes hyperboliques, algèbres d'opérateurs et un théorème de Jolissaint, C. R. Acad. Sci. Paris. Ser. I Math., 307 (1988), 771-774.   Google Scholar

[17]

V. Fischer and M. Ruzhansky, Sobolev spaces on graded Lie groups, Ann. Inst. Fourier (Grenoble), 67 (2017), 1671-1723.   Google Scholar

[18]

G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207.  doi: 10.1007/BF02386204.  Google Scholar

[19]

U. FranzG. HongF. LemeuxM. Ulrich and H. Zhang, Hypercontractivity of heat semigroups on free quantum groups, J. Oper. Theory, 77 (2017), 61-76.  doi: 10.7900/jot.2015nov13.2126.  Google Scholar

[20]

A. Freslon and R. Vergnioux, The radial MASA in free orthogonal quantum groups, J. Funct. Anal., 271 (2016), 2776-2807.  doi: 10.1016/j.jfa.2016.08.007.  Google Scholar

[21]

S. Giulini and G. Travaglini, $L^p$-estimates for matrix coefficients of irreducible representations of compact groups, Proc. Amer. Math. Soc., 80 (1980), 448-450.  doi: 10.2307/2043737.  Google Scholar

[22]

A. González-PérezM. Junge and J. Parcet, Smooth Fourier multipliers in group algebras via Sobolev dimension, Ann. Sci. Ec. Norm. Super., 50 (2017), 879-925.  doi: 10.24033/asens.2334.  Google Scholar

[23]

U. Haagerup, An example of a nonnuclear $C^*$-algebra, which has the metric approximation property, Invent. Math., 50 (1978/79), 279-293.  doi: 10.1007/BF01410082.  Google Scholar

[24]

Y. Isono, Examples of factors which have no Cartan subalgebras, Trans. Amer. Math. Soc., 367 (2015), 7917-7937.  doi: 10.1090/tran/6321.  Google Scholar

[25]

Y. Isono, Some prime factorization results for free quantum group factors, J. Reine Angew. Math., 722 (2017), 215-250.  doi: 10.1515/crelle-2014-0056.  Google Scholar

[26]

Y. Jiao and M. Wang, Noncommutative harmonic analysis on semigroups, Indiana Univ. Math. J., 66 (2017), 401-417.  doi: 10.1512/iumj.2017.66.6020.  Google Scholar

[27]

M. Junge and T. Mei, Noncommutative Riesz transforms-a probabilistic approach, Amer. J. Math., 132 (2010), 611-680.  doi: 10.1353/ajm.0.0122.  Google Scholar

[28]

M. Junge and T. Mei, BMO spaces associated with semigroups of operators, Math. Ann., 352 (2012), 691-743.  doi: 10.1007/s00208-011-0657-0.  Google Scholar

[29]

M. Junge, T. Mei and J. Parcet, An invitation to harmonic analysis associated with semigroups of operators, In Harmonic analysis and partial differential equations, Contemp. Math., Vol. 612, American Mathematical Society, Providence, RI, (2014), 107–122. Google Scholar

[30]

M. JungeT. Mei and J. Parcet, Smooth Fourier multipliers on group von Neumann algebras, Geom. Funct. Anal., 24 (2014), 1913-1980.  doi: 10.1007/s00039-014-0307-2.  Google Scholar

[31]

M. Junge, C. Palazuelos, J. Parcet and M. Perrin, Hypercontractivity in group von Neumann algebras, Mem. Amer. Math. Soc., 249 (2017), xii+83. doi: 10.1090/memo/1183.  Google Scholar

[32]

J. Kustermans and S. Vaes, Locally compact quantum groups, Ann. Sci. Ec. Norm. Super., 33 (2000), 837-934.  doi: 10.1016/S0012-9593(00)01055-7.  Google Scholar

[33]

J. Kustermans and S. Vaes, Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand., 92 (2003), 68-92.  doi: 10.7146/math.scand.a-14394.  Google Scholar

[34]

T. Mei and M. de la Salle, Complete boundedness of heat semigroups on the von Neumann algebra of hyperbolic groups, Trans. Amer. Math. Soc., 369 (2017), 5601-5622.  doi: 10.1090/tran/6825.  Google Scholar

[35]

T. Mei and Éric Ricard, Free Hilbert transforms, Duke Math. J., 166 (2017), 2153-2182.  doi: 10.1215/00127094-2017-0007.  Google Scholar

[36]

B. Nica, On the degree of rapid decay, Proc. Amer. Math. Soc., 138 (2010), 2341-2347.  doi: 10.1090/S0002-9939-10-10289-5.  Google Scholar

[37]

B. Nica, On operator norms for hyperbolic groups, J. Topol. Anal., 9 (2017), 291-296.  doi: 10.1142/S179352531750008X.  Google Scholar

[38] G. Pisier, Introduction to Operator Space Theory, London Mathematical Society Lecture Note Series, Vol. 294, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9781107360235.  Google Scholar
[39]

G. Pisier and Q. Xu, Non-commutative $L^p$-spaces, In Handbook of the Geometry of Banach Spaces, Vol. 2, North-Holland, Amsterdam, (2003), 1459–1517. Google Scholar

[40]

Éric Ricard and Q. Xu, Complex interpolation of weighted noncommutative $L_p$-spaces, Houston J. Math., 37 (2011), 1165-1179.   Google Scholar

[41] E. M.Stein, Topics in Harmonic Analysis Related to The Littlewood-Paley Theory, Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N. J., University of Tokyo Press, Tokyo, 1970.   Google Scholar
[42]

T. Timmermann, An invitation to Quantum Groups and Duality, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2008. From Hopf algebras to multiplicative unitaries and beyond. doi: 10.4171/043.  Google Scholar

[43]

A. V. Daele and S. Wang, Universal quantum groups, Int. J. Math., 7 (1996), 255-263.  doi: 10.1142/S0129167X96000153.  Google Scholar

[44]

N. Th. Varopoulos, Hardy-Littlewood theory for semigroups, J. Funct. Anal., 63 (1985), 240-260.  doi: 10.1016/0022-1236(85)90087-4.  Google Scholar

[45]

N. Th. Varopoulos, Analysis on Lie groups, J. Funct. Anal., 76 (1988), 346-410.  doi: 10.1016/0022-1236(88)90041-9.  Google Scholar

[46] N. Th. VaropoulosL. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, Cambridge Tracts in Mathematics, Vol. 100, Cambridge University Press, Cambridge, 1992.   Google Scholar
[47]

R. Vergnioux, The property of rapid decay for discrete quantum groups, J. Oper. Theory, 57 (2007), 303-324.   Google Scholar

[48]

R. Vergnioux and C. Voigt, The $K$-theory of free quantum groups, Math. Ann., 357 (2013), 355-400.  doi: 10.1007/s00208-013-0902-9.  Google Scholar

[49]

C. Voigt, The Baum-Connes conjecture for free orthogonal quantum groups, Adv. Math., 227 (2011), 1873-1913.  doi: 10.1016/j.aim.2011.04.008.  Google Scholar

[50]

N. R. Wallach, Harmonic Analysis on Homogeneous Spaces, Pure and Applied Mathematics, No. 19, Marcel Dekker, Inc., New York, 1973.  Google Scholar

[51]

S. Wang, Free products of compact quantum groups, Commun. Math. Phys., 167 (1995), 671-692.   Google Scholar

[52]

S. Wang, Quantum symmetry groups of finite spaces, Commun. Math. Phys., 195 (1998), 195-211.  doi: 10.1007/s002200050385.  Google Scholar

[53]

S. Wang, Lacunary Fourier series for compact quantum groups, Commun. Math. Phys., 349 (2017), 895-945.  doi: 10.1007/s00220-016-2670-3.  Google Scholar

[54]

S. L. Woronowicz, Compact matrix pseudogroups, Commun. Math. Phys., 111 (1987), 613-665.   Google Scholar

[55]

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