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Nondoubling Calderón–Zygmund theory: a dyadic approach

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Abstract

Given a measure \(\mu \) of polynomial growth, we refine a deep result by David and Mattila to construct an atomic martingale filtration of \(\mathrm {supp}(\mu )\) which provides the right framework for a dyadic form of nondoubling harmonic analysis. Despite this filtration being highly irregular, its atoms are comparable to balls in the given metric—which in turn are all doubling—and satisfy a weaker but crucial form of regularity. Our dyadic formulation is effective to address three basic questions:

  1. (i)

    A dyadic form of Tolsa’s RBMO space which contains it.

  2. (ii)

    Lerner’s domination and \(A_2\)-type bounds for nondoubling measures.

  3. (iii)

    A noncommutative form of nonhomogeneous Calderón–Zygmund theory.

Our martingale RBMO space preserves the crucial properties of Tolsa’s original definition and reveals its interpolation behavior with the \(L_p\) scale in the category of Banach spaces, unknown so far. On the other hand, due to some known obstructions for Haar shifts and related concepts over nondoubling measures, our pointwise domination theorem via sparsity naturally deviates from its doubling analogue. In a different direction, matrix-valued harmonic analysis over noncommutative \(L_p\) spaces has recently produced profound applications. Our analogue for nondoubling measures was expected for quite some time. Finally, we also find a dyadic form of the Calderón–Zygmund decomposition which unifies those by Tolsa and López-Sánchez/Martell/Parcet.

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References

  1. Caspers, M., Potapov, D., Sukochev, F., Zanin, D.: Weak type commutator and lipschitz estimates: resolution of the Nazarov–Peller conjecture. Preprint arXiv:1506.00778 [math.FA]

  2. Conde, J.M.: A note on dyadic coverings and nondoubling Calderón–Zygmund theory. J. Math. Anal. Appl. 397(2), 785–790 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Conde-Alonso, J.M., Parcet, J.: Atomic blocks for noncommutative martingales. Indiana Univ. Math. J. 65(4), 1425–1443 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Conde-Alonso, J.M., Rey, G.: On a pointwise estimate for positive dyadic shifts and some applications. Math. Ann. 365(3), 1111–1135 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Conde-Alonso, J.M., Mei, T., Parcet, J.: Large BMO spaces vs interpolation. Anal. PDE 8(3), 713–746 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cruz-Uribe, D., Martell, J.M., Pérez, C.: Sharp weighted estimates for classical operators. Adv. Math. 229(1), 408–441 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. David, G., Mattila, P.: Removable sets for Lipschitz harmonic functions in the plane. Rev. Mat. Iberoam. 16(1), 137–215 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Garnett, J.B., Jones, P.W.: \(\text{ BMO }\) from dyadic\(\text{ BMO }\). Pacif. J. Math. 99(2), 351–371 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  9. Garsia, A.M.: Martingale inequalities: Seminar notes on recent progress. In: Mathematics Lecture Notes Series. W. A. Benjamin, Inc., Reading/London/Amsterdam (1973)

  10. Hänninen, T.S.: Remark on dyadic pointwise domination and median oscillation decomposition. Houston J. Math. 43(1), 183–197 (2017)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Hytönen, T.P.: A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa. Publ. Mat. 54(2), 485–504 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hytönen, T.P.: The sharp weighted bound for general Calderón–Zygmund operators. Ann. of Math. (2) 175(3), 1473–1506 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Junge, M., Musat, M.: A noncommutative version of the John–Nirenberg theorem. Trans. Am. Math. Soc. 359(1), 115–142 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Lacey, M.T.: An elementary proof of the \({\rm A}_2\) bound. Israel J. Math. 217, 181–195 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lance, E.C.: Hilbert \(C^*\)-modules: a toolkit for operator algebraists. In: Proceedings of the London Mathematical Society Lecture Note Series, vol. 210. Cambridge University Press, Cambridge (1995)

  18. Lerner, A.K.: On pointwise estimates involving sparse operators. New York J. Math. 22, 341–349 (2016)

    MathSciNet  MATH  Google Scholar 

  19. Lerner, A.K.: A pointwise estimate for the local sharp maximal function with applications to singular integrals. Bull. Lond. Math. Soc. 42(5), 843–856 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lerner, A.K.: On an estimate of Calderón–Zygmund operators by dyadic positive operators. J. Anal. Math. 121, 141–161 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lerner, A.K., Nazarov, F.: Intuitive dyadic calculus: the basics. Preprint arXiv:1508.05639

  22. López-Sánchez, L.D., Martell, J.M., Parcet, J.: Dyadic harmonic analysis beyond doubling measures. Adv. Math. 267, 44–93 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mei, T.: Operator valued Hardy spaces. Mem. Am. Math. Soc 188(881), vi+64 (2007)

    MathSciNet  MATH  Google Scholar 

  24. Musat, M.: Interpolation between non-commutative BMO and non-commutative \(L_p\)-spaces. J. Funct. Anal. 202(1), 195–225 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Nazarov, F., Treil, S., Volberg, A.: Weak type estimates and Cotlar inequalities for Calderón–Zygmund operators on nonhomogeneous spaces. Int. Math. Res. Not. 9, 463–487 (1998)

    Article  MATH  Google Scholar 

  26. Nazarov, F., Treil, S., Volberg, A.: The \(Tb\)-theorem on non-homogeneous spaces. Acta Math. 190(2), 151–239 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Petermichl, S.: Of some sharp estimates involving Hilbert transform. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.), Michigan State University (2000)

  29. Pisier, G., Xu, Q.: Non-commutative martingale inequalities. Commun. Math. Phys. 189(3), 667–698 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. Tolsa, X.: BMO, \(\text{ H }^1\), and Calderón–Zygmund operators for non doubling measures. Math. Ann. 319(1), 89–149 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Tolsa, X.: A proof of the weak \((1,1)\) inequality for singular integrals with non doubling measures based on a Calderón–Zygmund decomposition. Publ. Mat. 45(1), 163–174 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. Tolsa, X.: Weighted norm inequalities for Calderón–Zygmund operators without doubling conditions. Publ. Mat. 51(2), 397–456 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Volberg, A., Zorin-Kranich, P.: Sparse domination on non-homogeneous spaces with an application to \({A}_p\) weights. Preprint arXiv:1606.03340

  34. Volberg, A.L., Eiderman, V.Y.: Nonhomogeneous harmonic analysis: 16 years of development. Uspekhi Mat. Nauk. 68(6), 3–58 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank Xavier Tolsa for bringing to our attention David and Mattila’s construction and for several fruitful discussions on the content of this paper. Both authors were partially supported by CSIC Project PIE 201650E030 and also by ICMAT Severo Ochoa Grant SEV-2015-0554, and the first named author was supported in part by ERC Grant 32501.

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Authors and Affiliations

  1. Department of Mathematics, Brown University, 151 Thayer Street, Providence, 02903, RI, USA

    José M. Conde-Alonso

  2. Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera 13-15, 28049, Madrid, Spain

    Javier Parcet

Authors
  1. José M. Conde-Alonso
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Correspondence to José M. Conde-Alonso.

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Communicated by Dachun Yang.

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Conde-Alonso, J.M., Parcet, J. Nondoubling Calderón–Zygmund theory: a dyadic approach. J Fourier Anal Appl 25, 1267–1292 (2019). https://doi.org/10.1007/s00041-018-9624-4

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  • DOI: https://doi.org/10.1007/s00041-018-9624-4

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