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Paley’s Inequality for Discrete Groups

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Abstract

This article studies Paley’s theory of lacunary Fourier series for von Neumann algebra of discrete groups. The results unify and generalize the work of Rudin (Fourier Analysis on Groups, Reprint of the 1962 original. Wiley Classics Library, A Wiley-Interscience Publication, Wiley, New York, 1990, Section 8) for abelian discrete ordered groups and the work of Lust-Piquard and Pisier (Ark Mat 29(2):241–260, 1991) for operator valued functions, and provide new examples of Paley sets and \(\Lambda (p)\) sets on free groups.

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Notes

  1. One can check directly that H satisfies the so-called Cotlar’s identity. So its p-boundedness follows from the classical iteration and interpolation argument. See [44, Lemma 8.5] for the details.

  2. One can see the conditional negativity of \(\psi _z\) by identifying \({\mathbb {F}}_2\) as a subgroup of the direct product \({\mathbb {F}}_2\times {\mathbb {Z}}^2\) via the group homomorphism

    $$\begin{aligned} g:\mapsto \left( g, \sum _{i=1}^N j_i, \sum _{i=1}^N k_i\right) . \end{aligned}$$

References

  1. Arveson, W.B.: Analyticity in operator algebras. Am. J. Math. 89, 578–642 (1967)

    Article  MathSciNet  Google Scholar 

  2. Berg, C., Christensen, J., Ressel, P.: Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions. Graduate Text in Mathematics. Springer (1984)

  3. Bekjan, T., Ospanov, K.: Complex interpolation of Noncommutative hardy spaces associated with semifinite von Neumann algebras. Acta Math. Sci. Ser. B 40(1), 245–260 (2020)

    Article  MathSciNet  Google Scholar 

  4. Blasco, O., Pełczyński, A.: Theorems of Hardy and Paley for vector-valued analytic functions and related classes of Banach spaces. Trans. Am. Math. Soc. 323(1), 335–367 (1991)

    Article  MathSciNet  Google Scholar 

  5. Blecher, D., Labuschagne, L.: Von Neumann Algebraic \(H^p\) Theory. Function Spaces, Contemp. Math., vol. 435, pp. 89–114. Amer. Math. Soc., Providence (2007)

  6. Bożejko, M.: Positive definite functions on the free group and the noncommutative Riesz product. (Italian summary) Boll. Un. Mat. Ital. A (6) 5(1), 13–21 (1986)

  7. Cadilhac, L.: Noncommutative Khintchine inequalities in interpolation spaces of \(L^{p}\)-spaces. Adv. Math. 352, 265–296 (2019)

    Article  MathSciNet  Google Scholar 

  8. Caspers, M.: Harmonic analysis and BMO-spaces of free Araki-Woods factors. Studia Math. 246(1), 71–107 (2019)

    Article  MathSciNet  Google Scholar 

  9. Caspers, M., Junge, M., Sukochev, F., Zanin, D.: BMO-estimates for non-commutative vector valued Lipschitz functions. J. Funct. Anal. 278(3), 108317 (2020)

    Article  MathSciNet  Google Scholar 

  10. Chen, Z., Xu, Q., Yin, Z.: Harmonic analysis on quantum tori. Commun. Math. Phys. 322(3), 755–805 (2013)

    Article  MathSciNet  Google Scholar 

  11. Dicks, W., Warren, Šunić, Z.: Orders on trees and free products of left-ordered groups. Can. Math. Bull. 63(2), 335–347 (2020)

  12. Ferguson, T., Mei, T., Simanek, B.: \(H^\infty \)-calculus for semigroup generators on BMO. Adv. Math. 347, 408–441 (2019)

    Article  MathSciNet  Google Scholar 

  13. Haagerup, U.: An example of a nonnuclear \(C^*\)-algebra, which has the metric approximation property. Invent. Math. 50(3), 279–293 (1979)

    Article  MathSciNet  Google Scholar 

  14. Haagerup, U., Musat, M.: On the best constants in noncommutative Khintchine-type inequalities. J. Funct. Anal. 250(2), 588–624 (2007)

    Article  MathSciNet  Google Scholar 

  15. Haagerup, U., Pisier, G.: Bounded linear operators between C*-algebras. Duke Math. J. 71(3), 889–925 (1993)

    Article  MathSciNet  Google Scholar 

  16. Harcharras, A.: Fourier analysis, Schur multipliers on \(S^p\) and non-commutative \(\Lambda (p)\)-sets. Studia Math. 137(3), 203–260 (1999)

    Article  MathSciNet  Google Scholar 

  17. Hermiller, S., Šunić, Z.: No positive cone in a free product is regular. Int. J. Algebra Comput. 27(8), 1113–1120 (2017)

    Article  MathSciNet  Google Scholar 

  18. Hare, K.E., Yang, R.: Sidon sets are proportionally Sidon with small Sidon constants. Can. Math. Bull. 62(4), 798–809 (2019)

    Article  MathSciNet  Google Scholar 

  19. Junge, M., Le Merdy, C., Xu, Q.: \(H^\infty \) functional calculus and square functions on noncommutative \(L^p\)-spaces. Astèrisque No. 305 (2006)

  20. Junge, M., Mei, T.: BMO spaces associated with semigroups of operators. Math. Ann. 352(3), 691–743 (2012)

    Article  MathSciNet  Google Scholar 

  21. Junge, M., Mei, T., Parcet, J.: Smooth Fourier multipliers on group von Neumann algebras. Geom. Funct. Anal. 24(6), 1913–1980 (2014)

    Article  MathSciNet  Google Scholar 

  22. Junge, M., Mei, T., Parcet, J., Xu, Q.: Work in progress

  23. Junge, M., Mei, T., Parcet, J., Xia, R.: Algebraic Calderón-Zygmund theory. Adv. Math. 376 (2021), Paper No. 107443 (2021)

  24. Kochneff, E., Sagher, Y., Zhou, K.C.: BMO estimates for lacunary series. Ark. Mat. 28(2), 301–310 (1990)

    Article  MathSciNet  Google Scholar 

  25. Leliévre, H.: Espaces BMO, inégalités de Paley et multiplicateurs idempotents. Studia Math. 123(3), 249–274 (1997)

    Article  MathSciNet  Google Scholar 

  26. Lust-Piquard, F., Pisier, G.: Noncommutative Khintchine and Paley inequalities. Ark. Mat. 29(2), 241–260 (1991)

    Article  MathSciNet  Google Scholar 

  27. Mei, T.: Operator valued Hardy spaces. Mem. Am. Math. Soc. 188(881) (2007)

  28. Mei, T.: Tent spaces associated with semigroups of operators. J. Funct. Anal. 255, 3356–3406 (2008)

    Article  MathSciNet  Google Scholar 

  29. Marsalli, M.: Noncommutative \(H^2\) spaces. Proc. Am. Math. Soc. 125, 779–784 (1997)

    Article  MathSciNet  Google Scholar 

  30. Marsalli, M., West, G.: Noncommutative \(H_p\) spaces. J. Oper. Theory 40, 339–355 (1998)

    MATH  Google Scholar 

  31. Marsalli, M., West, G.: The dual of noncommutative \(H^1\). Indiana Univ. Math. J. 47(2), 489–500 (1998)

    Article  MathSciNet  Google Scholar 

  32. Nazarov, F., Pisier, G., Treil, S., Volberg, A.: Sharp estimates in vector Carleson imbedding theorem and for vector Paraproducts. J. Reine Angew. Math. 542, 147–171 (2002)

    MathSciNet  MATH  Google Scholar 

  33. Paley, R.E.A.C.: On the lacunary coefficients of power series. Ann. Math. 2(34), 615–616 (1933)

    Article  MathSciNet  Google Scholar 

  34. Parcet, J., Rogers, K.: Twisted Hilbert transforms vs Kakeya sets of directions. J. Reine Angew. Math. 710, 137–172 (2016)

    MathSciNet  MATH  Google Scholar 

  35. Petermichl, S.: Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol. C. R. Acad. Sci. Paris Sér. I Math. 330(6), 455–460 (2000)

  36. Pisier, G.: Interpolation between \(H^p\) spaces and noncommutative generalizations. I. Pac. J. Math. 155(2), 341–368 (1992)

    Article  Google Scholar 

  37. Pisier, G.: Multipliers and lacunary sets in non-amenable groups. Am. J. Math. 117(2), 337–376 (1995)

    Article  MathSciNet  Google Scholar 

  38. Pisier, G.: Non-commutative Vector Valued \(L^p\)-spaces and Completely \(p\)-summing Map. Astérisque, 247 (1998)

  39. Pisier, G.: Real interpolation between row and column spaces. Bull. Pol. Acad. Sci. Math. 59(3), 237–259 (2011)

    Article  MathSciNet  Google Scholar 

  40. Pisier, G.: Grothendieck’s theorem, past and present. Bull. Am. Math. Soc. (N.S.) 49(2), 237–323 (2012)

  41. Pisier, G.: Completely Sidon sets in discrete groups. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 20(3), 891–923 (2020)

  42. Pisier, G.: Completely Sidon sets in \(C^*\)-algebras. Monatsh. Math. 187(2), 357–374 (2018)

    Article  MathSciNet  Google Scholar 

  43. Pisier, G.: Spectral gap properties of the unitary groups: around Rider’s results on non-commutative Sidon sets. In: Baranov, A., Kisliakov, S., Nikolski, N. (eds.) 50 Years with Hardy Spaces. Operator Theory: Advances and Applications, vol. 261. Birkhauser, Cham (2018)

  44. Pisier, G., Xu, Q.: Noncommutative \(L_p\)-Spaces. Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1459–1517, North-Holland, Amsterdam (2003)

  45. Pisier, G., Ricard, E.: The non-commutative Khintchine inequalities for \(0 < p < 1\). J. Inst. Math. Jussieu 16(5), 1103–1123 (2017)

    Article  MathSciNet  Google Scholar 

  46. Ricard, E.: A Markov dilation for self-adjoint Schur multipliers. Proc. Am. Math. Soc. 136(12), 4365–4372 (2008)

    Article  MathSciNet  Google Scholar 

  47. Rudin, W.: Trigonometric series with gaps. J. Math. Mech. 9, 203–227 (1960)

    MathSciNet  MATH  Google Scholar 

  48. Rudin, W.: Fourier Analysis on Groups, Reprint of the 1962 original. Wiley Classics Library. A Wiley-Interscience Publication. Wiley, New York (1990)

  49. Vinogradov, A.A.: On the free product of ordered groups. Mat. Sbornik N.S. (Russian) 25(67), 163–168 (1949)

  50. Wang, S.: Lacunary Fourier series for compact quantum groups. Commun. Math. Phys. 349, 895–945 (2017)

    Article  MathSciNet  Google Scholar 

  51. Xu, Q.: On the maximality of subdiagonal algebras. J. Oper. Theory 54, 137–146 (2005)

    MathSciNet  MATH  Google Scholar 

  52. Zygmund, A.: On Fourier coefficients and transforms of functions of two variables. Studia Math. 189–201 (1974)

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Acknowledgements

The authors would like to thank anonymous referees for constructive comments that improved this paper. Chuah, Liu, and Mei are partially supported by NSF grant DMS 1700171. Han is partially supported by NSFC grant 11761067.

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Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH, 43210-1174, USA

    Chian Yeong Chuah

  2. College of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, China

    Yazhou Han

  3. Department of Mathematics, Baylor University, 1301 S University Parks Dr, Waco, TX, 76798, USA

    Zhen-Chuan Liu & Tao Mei

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  1. Chian Yeong Chuah
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Correspondence to Yazhou Han.

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Communicated by Oscar Blasco.

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Chuah, C.Y., Han, Y., Liu, ZC. et al. Paley’s Inequality for Discrete Groups. J Fourier Anal Appl 28, 77 (2022). https://doi.org/10.1007/s00041-022-09971-1

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