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Product BMO, Little BMO, and Riesz Commutators in the Bessel Setting

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Abstract

In this paper, we study the product BMO space, little bmo space, and their connections with the corresponding commutators associated with Bessel operators studied by Weinstein, Huber, and Muckenhoupt–Stein. We first prove that the product BMO space in the Bessel setting can be used to deduce the boundedness of the iterated commutators with the Bessel Riesz transforms. We next study the little \({\mathrm{bmo}}\) space in this Bessel setting and obtain the equivalent characterization of this space in terms of commutators, where the main tool that we develop is the characterization of the predual of little bmo and its weak factorizations. We further show that in analogy with the classical setting the little \({\mathrm{bmo}}\) space is a proper subspace of the product \({\mathrm{BMO}}\) space. These extend the previous related results studied by Cotlar–Sadosky and Ferguson–Sadosky on the bidisc to the Bessel setting, where the usual analyticity and Fourier transform do not apply.

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References

  1. Andersen, K.F., Kerman, R.A.: Weighted norm inequalities for generalized Hankel conjugate transformations. Studia Math. 71, 15–26 (1981/82)

  2. Auscher, P., Hytönen, T.: Orthonormal bases of regular wavelets in spaces of homogeneous type. Appl. Comput. Harmon. Anal. 34, 266–296 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Betancor, J.J., Chicco Ruiz, A., Fariña, J.C., Rodríguez-Mesa, L.: Maximal operators, Riesz transforms and Littlewood–Paley functions associated with Bessel operators on BMO. J. Math. Anal. Appl. 363, 310–326 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Betancor, J.J., Dziubański, J., Torrea, J.L.: On Hardy spaces associated with Bessel operators. J. Anal. Math. 107, 195–219 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Betancor, J.J., Fariña, J.C., Buraczewski, D., Martínez, T., Torrea, J.L.: Riesz transform related to Bessel operators. Proc. R. Soc. Edinb. Sect. A 137, 701–725 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Betancor, J.J., Fariña, J.C., Sanabria, A.: On Littlewood–Paley functions associated with Bessel operators. Glasg. Math. J. 51, 55–70 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Betancor, J.J., Harboure, E., Nowak, A., Viviani, B.: Mapping properties of functional operators in harmonic analysis related to Bessel operators. Studia Math. 197, 101–140 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bramanti, M., Cerutti, M.C.: Commutators of singular integrals on homogeneous spaces. Boll. Un. Mat. Ital. B (7) 10, 843–883 (1996)

    MathSciNet  MATH  Google Scholar 

  9. Chang, S.-Y.A., Fefferman, R.: A continuous version of duality of \(H^1\) with \({{\rm BMO}}\) on the bidisc. Ann. Math. 112, 179–201 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  10. Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  11. Coifman, R.R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. (2) 103, 611–635 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dalenc, L., Ou, Y.: Upper bound for multi-parameter iterated commutators. Publ. Mat. 60, 191–220 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Duong, X.T., Li, J., Mao, S., Wu, H., Yang, D.: Compactness of Riesz transform commutator associated with Bessel operators. J. Anal. Math. (in press)

  14. Duong, X.T., Li, J., Wick, B.D., Yang, D.: Hardy space via factorization, and BMO space via commutators in the Bessel setting. Indiana Univ. Math. J. (in press)

  15. Duong, X.T., Li, J., Wick, B.D., Yang, D.: Characterizations of product Hardy spaces in Bessel setting. arXiv:1606.03500

  16. Duong, X.T., Li, J., Wick, B.D., Yang, D.: Commutators, little BMO and weak factorization. Ann. l’Inst. Fourier. (in press)

  17. Ferguson, S.H., Lacey, M.T.: A characterization of product BMO by commutators. Acta Math. 189, 143–160 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ferguson, S.H., Sadosky, C.: Characterizations of bounded mean oscillation on the polydisk in terms of Hankel operators and Carleson measures. J. Anal. Math. 81, 239–267 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Grafakos, L.: Graduate Texts in Mathematics. Classical Fourier Analysis, vol. 249, 2nd edn. Springer, Berlin (2008)

    Google Scholar 

  20. Haimo, D.T.: Integral equations associated with Hankel convolutions. Trans. Am. Math. Soc. 116, 330–375 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  21. Han, Y., Müller, D., Yang, D.: Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type. Math. Nachr. 279, 1505–1537 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Han, Y., Müller, D., Yang, D.: A theory of Besov and Triebel–Lizorkin spaces on metric measure spaces modeled on Carnot–Carathéodory spaces. Abstr. Appl. Anal. Article ID 893409 (2008)

  23. Han, Y., Li, J., Lu, G.: Multiparameter Hardy space theory on Carnot–Carathéodory spaces and product spaces of homogeneous type. Trans. Am. Math. Soc. 365, 319–360 (2013)

    Article  MATH  Google Scholar 

  24. Han, Y., Li, J., Lin, C.-C.: Criterions of the \(L^2\) boundedness and sharp endpoint estimates for singular integral operators on product spaces of homogeneous type. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2015)

  25. Han, Y., Li, J., Ward, L.A.: Hardy space theory on spaces of homogeneous type via orthonormal wavelet bases. Appl. Comput. Harmon. Anal. (in press)

  26. Huber, A.: On the uniqueness of generalized axially symmetric potentials. Ann. Math. 60, 351–358 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hytönen, T.: The sharp weighted bound for general Calderón–Zygmund operators. Ann. Math. (2) 175, 1473–1506 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hytönen, T., Kairema, A.: Systems of dyadic cubes in a doubling metric space. Colloq. Math. 126, 1–33 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Journé, J.L.: Calderón–Zygmund Operators, Pseudodifferential Operators And the Cauchy Integral of Calderón. Lecture Notes in Mathematics, vol. 994. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  30. Kairema, A., Li, J., Pereyra, C., Ward, L.A.: Haar bases on quasi-metric measure spaces, and dyadic structure theorems for function spaces on product spaces of homogeneous type. J. Funct. Anal. 271, 1793–1843 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kerman, R.A.: Boundedness criteria for generalized Hankel conjugate transformations. Can. J. Math. 30, 147–153 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  32. Krantz, S.G., Li, S.-Y.: Boundedness and compactness of integral operators on spaces of homogeneous type and applications. I. J. Math. Anal. Appl. 258, 629–641 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lacey, M., Petermichl, S., Pipher, J., Wick, B.D.: Multiparameter Riesz commutators. Am. J. Math. 131, 731–769 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lacey, M., Petermichl, S., Pipher, J., Wick, B.D.: Iterated Riesz commutators: a simple proof of boundedness, Harmonic analysis and partial differential equations. Contemp. Math. 505, 171–178 (2010)

    Article  MATH  Google Scholar 

  35. Muckenhoupt, B., Stein, E.M.: Classical expansions and their relation to conjugate harmonic functions. Trans. Am. Math. Soc. 118, 17–92 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  36. Nazarov, F., Reznikov, A., Volberg, A.: The proof of \(A_2\) conjecture in a geometrically doubling metric space. Indiana Univ. Math. J. 62, 1503–1533 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Nazarov, F., Reznikov, A., Volberg, A.: A simple sharp weighted estimate of the dyadic shifts on metric spaces with geometric doubling. Int. Math. Res. Not. IMRN, pp. 3771–3789 (2013)

  38. Tao, T.: Dyadic product \(H^1\), BMO, and Carleson’s counterexample, short stories. http://www.math.ucla.edu/~tao/preprints/harmonic.html

  39. Uchiyama, A.: The factorization of \(H^p\) on the space of homogeneous type. Pac. J. Math. 92, 453–468 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  40. Villani, M.: Riesz transforms associated to Bessel operators. Ill. J. Math. 52, 77–89 (2008)

    MathSciNet  MATH  Google Scholar 

  41. Weinstein, A.: Discontinuous integrals and generalized potential theory. Trans. Am. Math. Soc. 63, 342–354 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  42. Yang, D., Yang, D.: Real-variable characterizations of Hardy spaces associated with Bessel operators. Anal. Appl. (Singap.) 9, 345–368 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

X. T. Duong and J. Li are supported by ARC DP 160100153. J. Li is also supported by Macquarie University New Staff Grant. B. D. Wick’s research is supported in part by National Science Foundation DMS Grants #1603246 and #1560955. D. Yang is supported by the NNSF of China (Grant No. 11571289). The authors would like to thank the referee for careful reading of the paper and for helpful comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Macquarie University, Sydney, NSW, 2019, Australia

    Xuan Thinh Duong & Ji Li

  2. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Yumeng Ou

  3. Department of Mathematics, Washington University – St. Louis, St. Louis, MO, 63130-4899, USA

    Brett D. Wick

  4. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

    Dongyong Yang

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  1. Xuan Thinh Duong
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Correspondence to Dongyong Yang.

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Duong, X.T., Li, J., Ou, Y. et al. Product BMO, Little BMO, and Riesz Commutators in the Bessel Setting. J Geom Anal 28, 2558–2601 (2018). https://doi.org/10.1007/s12220-017-9920-2

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  • DOI: https://doi.org/10.1007/s12220-017-9920-2

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