Skip to main content
Log in

Operator-Valued Triebel–Lizorkin Spaces

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

This paper is devoted to the study of operator-valued Triebel–Lizorkin spaces. We develop some Fourier multiplier theorems for square functions as our main tool, and then study the operator-valued Triebel–Lizorkin spaces on \(\mathbb {R}^d\). As in the classical case, we connect these spaces with operator-valued local Hardy spaces via Bessel potentials. We show the lifting theorem, and get interpolation results for these spaces. We obtain Littlewood–Paley type, as well as the Lusin type square function characterizations in the general way. Finally, we establish smooth atomic decompositions for the operator-valued Triebel–Lizorkin spaces. These atomic decompositions play a key role in our recent study of mapping properties of pseudo-differential operators with operator-valued symbols.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bekjan, T., Chen, Z., Perrin, M., Yin, Z.: Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales. J. Funct. Anal. 258(7), 2483–2505 (2010)

    Article  MathSciNet  Google Scholar 

  2. Bergh, J., Löfström, J.: Interpolation Spaces: An Introduction. Springer, Berlin (1976)

    Book  Google Scholar 

  3. Coifman, R., Meyer, Y., Stein, E.M.: Some new function spaces and their applications to Harmonic analysis. J. Funct. Anal. 62(2), 304–335 (1985)

    Article  MathSciNet  Google Scholar 

  4. Frazier, M., Jawerth, B.: The \(\varphi \)-transform and applications to distribution spaces. In: Cwikel, M., et al. (eds.) Function Spaces and Applications, Springer Lecture Notes in Mathematics, vol. 1302, pp. 223–246 (1988)

  5. Frazier, M., Jawerth, B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93(1), 34–170 (1990)

    Article  MathSciNet  Google Scholar 

  6. Goldberg, D.: A local version of real Hardy spaces. Duke Math. J. 46(1), 27–42 (1979)

    Article  MathSciNet  Google Scholar 

  7. Hong, G., López-Snchez, L.D., Martell, J.M., Parcet, J.: Calderón–Zygmund operator associated to matrix-valued kernels. Int. Math. Res. Not. 5, 1221–1252 (2014)

    Article  Google Scholar 

  8. Hong, G., Mei, T.: John–Nirenberg inequality and atomic decomposition for noncommutative martingales. J. Funct. Anal. 263(4), 1064–1097 (2012)

    Article  MathSciNet  Google Scholar 

  9. Jiao, Y., Sukochev, F., Zanin, D., Zhou, D.: Johnson–Schechtman inequalities for noncommutative martingales. J. Funct. Anal. 272(3), 976–1016 (2017)

    Article  MathSciNet  Google Scholar 

  10. Junge, M.: Doob’s inequality for non-commutative martingales. J. Reine Angew. Math. 549, 149–190 (2002)

    MathSciNet  MATH  Google Scholar 

  11. Junge, M., Le Merdy, C., Xu, Q.: \(H^{\infty }\)-Functional Calculus and Square Functions on Noncommutative \(L_{p}\)-Spaces. Astérisque, vol. 305 (2006)

  12. Junge, M., Xu, Q.: Non-commutative Burkholder/Rosenthal inequalities. I. Ann. Prob. 31(2), 948–995 (2003)

    Article  Google Scholar 

  13. Junge, M., Xu, Q.: Non-commutative Burkholder/Rosenthal inequalities. II. Isr. J. Math. 167, 227–282 (2008)

    Article  Google Scholar 

  14. Junge, M., Xu, Q.: Noncommutative maximal ergodic theorems. J. Am. Math. Soc. 20, 385–439 (2007)

    Article  MathSciNet  Google Scholar 

  15. Junge, M., Mei, T.: Noncommutative Riesz transforms—a probabilistic approach. Am. J. Math. 132, 611–681 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Mei, T.: Operator-valued Hardy Spaces. Memoirs of the American Mathematical Society, vol. 188, no. 881 (2007)

  19. Mei, T.: Tent spaces associated with semigroups of operators. J. Funct. Anal. 255(12), 3356–3406 (2008)

    Article  MathSciNet  Google Scholar 

  20. Parcet, J.: Pseudo-localization of singular integrals and noncommutative Calderón–Zygmund theory. J. Funct. Anal. 256(2), 509–593 (2009)

    Article  MathSciNet  Google Scholar 

  21. Parcet, J., Randrianantoanina, N.: Gundy’s decomposition for non-commutative martingales and applications. Proc. Lond. Math. Soc. 93, 227–252 (2006)

    Article  MathSciNet  Google Scholar 

  22. Pisier, G.: Noncommutative Vector Valued \(L_p\) Spaces and Completely \(p\)-Summing Maps. Astérisque, vol. 247 (1998)

  23. Pisier, G., Xu, Q.: Non-commutative martingale inequalities. Commun. Math. Phys. 189, 667–698 (1997)

    Article  MathSciNet  Google Scholar 

  24. Pisier, G., Xu, Q.: Noncommutative \(L_{p}\)-spaces. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1459–1517. North-Holland, Amsterdam (2003)

    Chapter  Google Scholar 

  25. Randrianantoanina, N.: Noncommutative martingale transforms. J. Funct. Anal. 194, 181–212 (2002)

    MathSciNet  MATH  Google Scholar 

  26. Randrianantoanina, N.: Conditional square functions for noncommutative martingales. Ann. Prob. 35, 1039–1070 (2007)

    Article  Google Scholar 

  27. Randrianantoanina, N., Wu, L.: Noncommutative Burkholder/Rosenthal inequalities associated with convex functions. Ann. Inst. Henri Poincaré Probab. Stat. 53(4), 1575–1605 (2017)

    Article  MathSciNet  Google Scholar 

  28. Randrianantoanina, N., Wu, L., Xu, Q.: Noncommutative Davis type decompositions and applications. J. Lond. Math. Soc. (2018). https://doi.org/10.1112/jlms.12166

    Article  Google Scholar 

  29. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30 Princeton University Press, Princeton (1970)

  30. Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory integrals. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton (1993)

  31. Triebel, H.: Theory of Function Spaces. Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel (2010)

    Google Scholar 

  32. Triebel, H.: Theory of Function Spaces. II. Monographs in Mathematics, vol. 84. Birkhäuser Verlag, Basel (1992)

  33. Xia, R., Xiong, X.: Operator-valued local Hardy spaces. to appear in J. Operator Th

  34. Xia, R., Xiong, X.: Pseudo-Differential Operators with Operator-Valued Kernel (2018). arXiv:1804.03435

  35. Xia, R., Xiong, X., Xu, Q.: Characterizations of operator-valued Hardy spaces and applications to harmonic analysis on quantum tori. Adv. Math. 291, 183–227 (2016)

    Article  MathSciNet  Google Scholar 

  36. Xiong, X., Xu, Q., Yin, Z.: Function spaces on quantum tori. C. R. Math. Acad. Sci. Paris 353(8), 729–734 (2015)

    Article  MathSciNet  Google Scholar 

  37. Xiong, X., Xu, Q., Yin, Z.: Sobolev, Besov and Triebel–Lizorkin Spaces on Quantum Tori. Memoirs of the American Mathematical Society, vol. 252, no. 1203 (2018)

  38. Xu, Q.: Noncommutative \(L_{p}\)-Spaces and Martingale Inequalities. Book manuscript (2007)

Download references

Acknowledgements

The authors are greatly indebted to Professor Quanhua Xu for having suggested to them the subject of this paper, for many helpful discussions and very careful reading of this paper. The authors are partially supported by the National Natural Science Foundation of China (Grant No. 11301401).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Runlian Xia.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xia, R., Xiong, X. Operator-Valued Triebel–Lizorkin Spaces. Integr. Equ. Oper. Theory 90, 65 (2018). https://doi.org/10.1007/s00020-018-2491-1

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00020-018-2491-1

Mathematics Subject Classification

Keywords

Navigation