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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[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.

[37]

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

[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.
[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.

[40]

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

[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. 
[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.

[43]

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

[44]

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

[45]

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

[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. 
[47]

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

[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.

[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.

[50]

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

[51]

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

[52]

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

[53]

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

[54]

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

[55]

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.

[56]

X. Xiong, Noncommutative harmonic analysis on semigroup and ultracontractivity, Indiana Univ. Math. J., 66 (2017), 1921-1947.  doi: 10.1512/iumj.2017.66.6221.

[57]

X. Xiong, Q. Xu and Z. Yin, Sobolev, Besov and Triebel-Lizorkin spaces on quantum tori, Mem. Amer. Math. Soc., 252 (2018), vi+118. doi: 10.1090/memo/1203.

[58]

Q. Xu, Interpolation of operator spaces, J. Funct. Anal., 139 (1996), 500-539.  doi: 10.1006/jfan.1996.0094.

[59]

S. Youn, Hardy-Littlewood inequalities on compact quantum groups of Kac type, Anal. Partial Differ. Equ., 11 (2018), 237-261.  doi: 10.2140/apde.2018.11.237.

[60]

S. Youn, Entropic uncertainty relations under localizations on discrete quantum groups, J. Math. Phys., 59 (2018), 073501, 21. doi: 10.1063/1.5037820.

[61]

S. Youn, Multipliers and The Similarity Property for Topological Quantum Groups, Ph.D thesis, Graduate School of Seoul National University, 2018.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[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.

[37]

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

[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.
[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.

[40]

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

[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. 
[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.

[43]

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

[44]

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

[45]

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

[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. 
[47]

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

[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.

[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.

[50]

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

[51]

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

[52]

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

[53]

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

[54]

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

[55]

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.

[56]

X. Xiong, Noncommutative harmonic analysis on semigroup and ultracontractivity, Indiana Univ. Math. J., 66 (2017), 1921-1947.  doi: 10.1512/iumj.2017.66.6221.

[57]

X. Xiong, Q. Xu and Z. Yin, Sobolev, Besov and Triebel-Lizorkin spaces on quantum tori, Mem. Amer. Math. Soc., 252 (2018), vi+118. doi: 10.1090/memo/1203.

[58]

Q. Xu, Interpolation of operator spaces, J. Funct. Anal., 139 (1996), 500-539.  doi: 10.1006/jfan.1996.0094.

[59]

S. Youn, Hardy-Littlewood inequalities on compact quantum groups of Kac type, Anal. Partial Differ. Equ., 11 (2018), 237-261.  doi: 10.2140/apde.2018.11.237.

[60]

S. Youn, Entropic uncertainty relations under localizations on discrete quantum groups, J. Math. Phys., 59 (2018), 073501, 21. doi: 10.1063/1.5037820.

[61]

S. Youn, Multipliers and The Similarity Property for Topological Quantum Groups, Ph.D thesis, Graduate School of Seoul National University, 2018.

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