Skip to main content
SpringerLink
Log in

Sparse and weighted estimates for generalized Hörmander operators and commutators

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

In this paper a pointwise sparse domination for generalized Hörmander and also for iterated commutators with those operators is provided generalizing the sparse domination result in Lerner et al. (Adv Math 319:153–181, 2017). Relying upon that sparse domination a number of quantitative estimates are derived. Some of them are improvements and complementary results to those contained in a series of papers due to Lorente et al. (Stud Math 195(2):157–192, 2009, J Math Anal Appl 342(2):1399–1425, 2008, J Fourier Anal Appl 11(5):497–509, 2005). Also the quantitative endpoint estimates in Lerner et al. (Adv Math 319:153–181, 2017) are extended to iterated commutators. Other results that are obtained in this work are some local exponential decay estimates for generalized Hörmander operators in the spirit of Ortiz-Caraballo et al. (Math Ann 357(4):1217–1243, 2013) and some negative results concerning Coifman–Fefferman estimates for a certain class of kernels satisfying particular generalized Hörmander conditions.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bui, T.A., Conde-Alonso, J.M., Duong, X.T., Hormozi, M.: A note on weighted bounds for singular operators with nonsmooth kernels. Stud. Math. 236(3), 245–269 (2017)

    Article  MathSciNet  Google Scholar 

  2. Caldarelli, M., Lerner, A.K., Ombrosi, S.: On a counterexample related to weighted weak type estimates for singular integrals. Proc. Amer. Math. Soc. 145(7), 3005–3012 (2015)

    Article  MathSciNet  Google Scholar 

  3. Chung, D., Pereyra, M.C., Perez, C.: Sharp bounds for general commutators on weighted Lebesgue spaces. Trans. Am. Math. Soc. 364(3), 1163–1177 (2012)

    Article  MathSciNet  Google Scholar 

  4. Coifman, R.R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. (2) 103(3), 611–635 (1976)

    Article  MathSciNet  Google Scholar 

  5. Conde-Alonso, J.M., Culiuc, A., Di Plinio, F., Yumeng, O.: A sparse domination principle for rough singular integrals. Anal. PDE 10(5), 1255–1284 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Cruz-Uribe, D., Martell, J.M., Pérez, C.: Extrapolation from \(A_\infty \) weights and applications. J. Funct. Anal. 213(2), 412–439 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Cruz-Uribe, D.V., Martell, J.M., Pérez, C.: Weights, extrapolation and the theory of Rubio de Francia. In: Gohberg, I., Krupnik, N. (eds.) Operator Theory: Advances and Applications, vol. 215. Springer Basel AG, Basel (2011)

    MATH  Google Scholar 

  10. Domingo-Salazar, C., Lacey, M.T., Rey, G.: Borderline weak-type estimates for singular integrals and square functions. Bull. Lond. Math. Soc. 48(1), 63–73 (2016)

    Article  MathSciNet  Google Scholar 

  11. Duoandikoetxea, J.: Fourier Analysis. Graduate Studies in Mathematics, vol. 29. American Mathematical Society, Providence, RI [Translated and revised from the 1995 Spanish original by David Cruz-Uribe (2001)]

  12. Fujii, N.: Weighted bounded mean oscillation and singular integrals. Math. Jpn. 22(5), 529–534 (1977/1978)

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

    Article  MathSciNet  Google Scholar 

  14. Hytönen, T.P.: The Holmes–Wick theorem on two-weight bounds for higher order commutators revisited. Arch. Math. (Basel) 107(4), 389–395 (2016)

    Article  MathSciNet  Google Scholar 

  15. Hytönen, T.P., Li, K.: Weak and strong \(A_p\)-\(A_\infty \) estimates for square functions and related operators. Proc. Amer. Math. Soc. 146(6), 2497–2507 (2018)

    Article  MathSciNet  Google Scholar 

  16. Hytönen, T.P., Pérez, C.: Sharp weighted bounds involving \(A_\infty \). Anal. PDE 6(4), 777–818 (2013)

    Article  MathSciNet  Google Scholar 

  17. Hytönen, T.P., Pérez, C.: The \(L(\log L)^\epsilon \) endpoint estimate for maximal singular integral operators. J. Math. Anal. Appl. 428(1), 605–626 (2015)

    Article  MathSciNet  Google Scholar 

  18. Hytönen, T.P., Pérez, C., Rela, E.: Sharp reverse Hölder property for \(A_\infty \) weights on spaces of homogeneous type. J. Funct. Anal. 263(12), 3883–3899 (2012)

    Article  MathSciNet  Google Scholar 

  19. Hytönen, T.P., Roncal, L., Tapiola, O.: Quantitative weighted estimates for rough homogeneous singular integrals. Israel J. Math. 218(1), 133–164 (2017)

    Article  MathSciNet  Google Scholar 

  20. Lacey, M.T.: An elementary proof of the \(A_2\) bound. Israel J. Math. 217(1), 181–195 (2017)

    Article  MathSciNet  Google Scholar 

  21. Lerner, A.K.: A simple proof of the \(A_2\) conjecture. Int. Math. Res. Not. IMRN 14, 3159–3170 (2013)

    Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  23. Lerner, A.K., Nazarov, F.: Intuitive dyadic calculus: the basics. Expo. Math. 37(3), 225–265 (2019)

    Article  MathSciNet  Google Scholar 

  24. Lerner, A.K., Ombrosi, S., Rivera-Rí os, I.P.: On pointwise and weighted estimates for commutators of Calderón–Zygmund operators. Adv. Math. 319, 153–181 (2017)

    Article  MathSciNet  Google Scholar 

  25. Li, K.: Sparse domination theorem for multilinear singular integral operators with \(L^{r}\)-Hörmander condition. Michigan Math. J. 67(2), 253–265 (2018)

    Article  MathSciNet  Google Scholar 

  26. Liu, L., Luque, T.: A \(B_p\) condition for the strong maximal function. Trans. Am. Math. Soc. 366(11), 5707–5726 (2014)

    Article  Google Scholar 

  27. Lorente, M., Martell, J.M., Pérez, C., Riveros, M.S.: Generalized Hörmander conditions and weighted endpoint estimates. Stud. Math. 195(2), 157–192 (2009)

    Article  Google Scholar 

  28. Lorente, M., Martell, J.M., Riveros, M.S., de la Torre, A.: Generalized Hörmander’s conditions, commutators and weights. J. Math. Anal. Appl. 342(2), 1399–1425 (2008)

    Article  MathSciNet  Google Scholar 

  29. Lorente, M., Riveros, M.S., de la Torre, A.: Weighted estimates for singular integral operators satisfying Hörmander’s conditions of Young type. J. Fourier Anal. Appl. 11(5), 497–509 (2005)

    Article  MathSciNet  Google Scholar 

  30. Martell, J.M., Pérez, C., Trujillo-González, R.: Lack of natural weighted estimates for some singular integral operators. Trans. Am. Math. Soc. 357(1), 385–396 (2005)

    Article  MathSciNet  Google Scholar 

  31. O’Neil, R.: Fractional integration in Orlicz spaces. I. Trans. Am. Math. Soc. 115, 300–328 (1965)

    Article  MathSciNet  Google Scholar 

  32. Ortiz-Caraballo, C.: Conmutadores de integrales singulares y pesos \(A_{1}\). Ph.D. thesis, Universidad de Sevilla (2011)

  33. Ortiz-Caraballo, C., Pérez, C., Rela, E.: Exponential decay estimates for singular integral operators. Math. Ann. 357(4), 1217–1243 (2013)

    Article  MathSciNet  Google Scholar 

  34. Pérez, C.: Endpoint estimates for commutators of singular integral operators. J. Funct. Anal. 128(1), 163–185 (1995)

    Article  MathSciNet  Google Scholar 

  35. Pérez, C.: On sufficient conditions for the boundedness of the Hardy–Littlewood maximal operator between weighted \(L^p\)-spaces with different weights. Proc. Lond. Math. Soc. (3) 71(1), 135–157 (1995)

    Article  Google Scholar 

  36. Pérez, C., Pradolini, G.: Sharp weighted endpoint estimates for commutators of singular integrals. Mich. Math. J. 49(1), 23–37 (2001)

    Article  MathSciNet  Google Scholar 

  37. Pérez, C., Rivera-Ríos, I.P.: Three observations on commutators of singular integral operators with BMO functions. Harmonic analysis, partial dierential equations, Banach spaces, and operator theory. Vol. 2, 287–304, Assoc. Women Math. Ser., 5, Springer, Cham, (2017)

    Google Scholar 

  38. Pérez, C., Rivera-Ríos, I.P.: Borderline weighted estimates for commutators of singular integrals. Isr. J. Math. 217(1), 435–475 (2017)

    Article  MathSciNet  Google Scholar 

  39. Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Monographs and Textbooks in Pure and Applied Mathematics, vol. 146. Marcel Dekker Inc, New York (1991)

    Google Scholar 

  40. Wilson, J.M.: Weighted inequalities for the dyadic square function without dyadic \(A_\infty \). Duke Math. J. 55(1), 19–50 (1987)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The first author would like Carlos Pérez for inviting him to visit BCAM between January and April 2017, and BCAM for the warm hospitality shown during his visit.

Author information

Authors and Affiliations

  1. BCAM - Basque Center for Applied Mathematics, Bilbo, Spain

    Gonzalo H. Ibañez-Firnkorn & Israel P. Rivera-Ríos

  2. FaMAF, Universidad Nacional de Córdoba, Córdoba, Argentina

    Gonzalo H. Ibañez-Firnkorn

  3. CIEM-CONICET, Córdoba, Argentina

    Gonzalo H. Ibañez-Firnkorn

  4. Departamento de Matemáticas/Matematika Saila, Universidad del País Vasco/Euskal Herriko Unibertsitatea, Leioa, Spain

    Israel P. Rivera-Ríos

  5. Departamento de Matemática, Universidad Nacional del Sur, Bahía Blanca, Argentina

    Israel P. Rivera-Ríos

  6. INMABB-CONICET, Bahía Blanca, Argentina

    Israel P. Rivera-Ríos

Authors
  1. Gonzalo H. Ibañez-Firnkorn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Israel P. Rivera-Ríos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Israel P. Rivera-Ríos.

Additional information

Communicated by Karlheinz Gröchenig.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Gonzalo H. Ibáñez-Firnkorn: Supported by CONICET and SECYT-UNC and also by the Basque Government through the BERC 2014–2017 program and by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323. Israel P. Rivera-Ríos: Supported by the Basque Government through the BERC 2014–2017 program and by the Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2013-0323 and also through the Projects MTM2014-53850-P and MTM2012-30748.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ibañez-Firnkorn, G.H., Rivera-Ríos, I.P. Sparse and weighted estimates for generalized Hörmander operators and commutators. Monatsh Math 191, 125–173 (2020). https://doi.org/10.1007/s00605-019-01349-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-019-01349-8

Keywords

Mathematics Subject Classification

Search

Navigation