Skip to main content
SpringerLink
Log in

Some estimates of multilinear operators on weighted amalgam spaces \((L^p,L^q_w)_t(\mathbb R^n)\)

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

We introduce the weighted amalgam space \((L^p,L^q_w)_t(\mathbb R^n)\), where \(t\in(0,\infty)\), \(p,q\in(1,\infty]\) and \(w\) is a weight. We establish the mapping properties of the multilinear Hardy-Littlewood maximal operator and the sparse operator on weighted amalgam spaces as well as the weighted boundedness of some classical multilinear operators in harmonic analysis over these spaces, such as the multilinear Calderòn-Zygmund operator, the multilinear singular integral operator with the non-smooth kernel, the multilinear Littlewood-Paley \(g\) function, the multilinear Marcinkiewicz integral and the multilinear fractional integral operator.

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. A. Amenta, Interpolation and embeddings of weighted tent spaces, J. Fourier Anal. Appl., 24 (2018), 108–140.

  2. P. Auscher and C. Prisuelos-Arribas, Tent space boundedness via extrapolation, Math. Z., 286 (2017), 1575-1604.

  3. P. Auscher and M. Mourgoglou, Representation and uniqueness for boundary value elliptic problems via first order systems, Rev. Mat. Iberoam, 35 (2019), 241– 315.

  4. A. Benedek and R. Panzone, The spaces Lp, with mixed norm, Duke Math. J., 28 (1961), 301–324.

  5. J. C. Chen and G. E. Hu, Weighted vector-valued bounds for a class of multilinear singular integral operators and applications, J. Korean Math. Soc., 55 (2018), 671–694.

  6. J. Cohen, A sharp estimate for a multilinear singular integral on \(\mathbb{R}^{n}\), Indiana Univ. Math. J., 30 (1981), 693–702.

  7. J. Cohen, Multilinear singular integrals, Thesis, Washington University (St. Louis, MO, 1997).

  8. J. Cohen and J. Gosselin, On multilinear singular integral operators on \(\mathbb{R}^{n}\), Studia Math., 72 (1982), 199–223.

  9. R. Coifman and Y. Meyer, On commutators of singular integral and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315–331.

  10. R. Coifman and Y. Meyer, Au delá des opérateurs pseudo-différentiels, Astérisque 57, Soc. Math. France (Paris, 1978).

  11. R. Coifman and Y. Meyer, Ondelettes et Opérateurs, III, Hermann (Paris, 1990).

  12. W. Damían, M. Hormoz and K. Li, New bounds for bilinear Calderón–Zygmund operators and applications, Rev. Mat. Iberoam., 34 (2018), 1177–1210.

  13. W. Damían, A. K. Lerner and C. Pérez, Sharp weighted bounds for multilinear maximalfunctions and Calderón–Zygmund operators, J. Fourier Anal. Appl., 21 (2015), 161–181.

  14. X. Duong, L. Grafakos and L. Yan, Multilinear operators with non-smooth kernels and commutators of singular integrals, Trans. Amer. Math. Soc., 362 (2010), 2089–2113.

  15. H. G. Feichtinger, Banach convolution algebras of Wiener type, in: Functions, Series, Operators, Proc. Conf. Budapest 38, Colloq. Math. Soc. János Bolyai (1980), pp. 509–524.

  16. J. J. F. Fournier and J. Stewart, Amalgams of Lp and \(\mathcal{L}\)q, Bull. Amer. Math. Soc. (N.S.), 13 (1985), 1–21.

  17. L. Grafakos, On multilinear fractional integrals, Studia Math., 102 (1992), 49–56.

  18. L. Grafakos and N. Kalton, Multilinear Calderón–Zygmund operators on Hardy spaces, Collect. Math., 52 (2001), 169–179.

  19. L. Grafakos and R. H. Torres, Multilinear Calderón–Zygmund theory, Adv. Math., 165 (2002), 124–164.

  20. L. Grafakos and R. H. Torres, Maximal operator and weighted norm inequalities for multilinear singular integrals, Indiana Univ. Math. J., 51 (2002), 1261–1276.

  21. L. Grafakos and R. H. Torres, On multilinear singular integrals of Calderón–Zygmund type, in: Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, Publ. Mat., Extra Vol (2002), pp. 57–91.

  22. C. Heil, Wiener amalgam spaces in generalized harmonic analysis and wavelet theory, Ph.D. thesis, University of Maryland (CollegePark, MD, 1990).

  23. C. Heil, An introduction to weighted Wiener amalgams, in: Wavelets and their Applications, edited by M. Krishna, R. Radha, and S. Thangavelu, Allied Publishers (New Delhi, 2003), pp. 183–216.

  24. K.-P. Ho, Operators on Orlicz-slice spaces and Orlicz-slice Hardy spaces, J. Math. Anal. Appl., 503 (2021), 125279.

  25. F. Holland, Harmonic analysis on amalgams of Lp and \(\mathcal{L}\)q, J. London Math. Soc., 10 (1975), 295–305.

  26. C. E. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett., 6 (1999), 1–15.

  27. V. Kokilashvili, M. Mastyło and A. Meskhi, On the boundedness of multilinear fractional integral operators, J. Geom. Anal., 30 (2020), 667–679.

  28. A. K. Lerner, A simple proof of the A2 conjecture, Int. Math. Res. Not., 14 (2013), 3159–3170.

  29. A. K. Lerner, S. Ombrossi, C. Pérez, R. H. Torres and R. Trojillo-Gonzalez, New maximal functions and multiple weights for the multilinear Calderón–Zygmund theorey, Adv. Math., 220 (2009), 1222–1264.

  30. K. W. Li, Sparse domination theorem for multilinear singular integral operators with Lr-Hörmander condition, Mich. Math. J., 67 (2016), 253–265.

  31. K. W. Li, K. Moen and W. C. Sun, The sharp weighted bound for multilinear maximal functions and Calderón–Zygmund operators, J. Fourier Anal. Appl., 20 (2014), 751–765.

  32. J. E. Littlewood and R. E. A. C. Paley, Theorems on Fourier series and power series, J. London Math. Soc., 6 (1931), 230–233.

  33. J. E. Littlewood and R. E. A. C. Paley, Theorems on Fourier series and power series. II, Proc. Lond. Math. Soc, 42 (1937), 52–89.

  34. J. E. Littlewood and R. E. A. C. Paley, Theorems on Fourier series and power series. III, Proc. Lond. Math. Soc., 43 (1938), 105–126.

  35. Y. Lu, S. B. Wang and J. Zhou, Boundedness of some operators on weighted amalgam spaces, arXiv:2110.01193 (2021).

  36. K. Moen, Weighted inequalities for multilinear fractional integral operators, Collect. Math., 60 (2009), 213–238.

  37. B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207–226.

  38. S. G. Shi, Q. Y. Xue and K. Yabuta, On the boundedness of multilinear Littlewood– Paley \(\text{g}^{*}_{\lambda}\) function, J. Math. Pure. Appl., 101 (2014), 394–413.

  39. E. M. Stein, On the functions of Littlewood–Paley, Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc., 88 (1958), 430–466.

  40. E. M. Stein, On some function of Littlewood–Paley and Zygmund, Bull. Amer. Math. Soc., 67 (1961), 99–101.

  41. S. B.Wang,Weighted bounds for multilinear Marcinkiewicz integrals with non-smooth kernel, Acta Math. Sinica (Chin. Ser.), 61 (2018), 12 pp.

  42. N. Wiener, On the representation of functions by trigonometrical integrals, Math. Z., 24 (1926), 575–616.

  43. N. Wiener, Tauberian theorems, Ann. Math., 33 (1932), 1–100.

  44. L. R. Williams, Generalized Hausdorff inequalities and mixed-norm spaces, Pacific J. Math., 38 (1971), 823–833.

  45. D. Wu and D. Y. Yan, Sharp constants for a class of multilinear integral operators and some applications, Sci. China Math., 59 (2016), 907–920.

  46. C. Xi and Q. Y. Xue, Weighted estimates for a class of multilinear fractional type operators, J. Math. Anal. Appl., 362 (2010), 355–373.

  47. Q. Y. Xue, Y. Ding and K. Yabuta, Weighted estimate for a class of Littlewood–Paley operators, Taiwan J. Math., 11 (2007), 339–365.

  48. Q. Y. Xue, X. X. Peng and K. Yabuta, On the theory of multilinear Littlewood–Paley g-function, J. Math. Soc. Japan., 67 (2015), 535–559.

  49. Y. Y. Zhang, D. C. Yang, W. Yuan and S. B. Wang, Real-variable characterizations of Orlicz-slice Hardy spaces, Anal. Appl., 17 (2019), 597–664.

Download references

Acknowledgement

The authors would like to express their thanks to the referees for valuable advice regarding a previous version of this paper.

Author information

Authors and Affiliations

  1. College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, P.R. China

    Y. Lu & J. Zhou

  2. College of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing, 404130, P.R. China

    S. B. Wang

Authors
  1. Y. Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. B. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. B. Wang.

Additional information

SBW is supported by the Natural Science Foundation Project of Chongqing, China (Grant No. cstc2021jcyj-msxmX0705).

JZ is supported by the National Natural Science Foundation of China (Grant No. 12061069).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lu, Y., Wang, S.B. & Zhou, J. Some estimates of multilinear operators on weighted amalgam spaces \((L^p,L^q_w)_t(\mathbb R^n)\). Acta Math. Hungar. 168, 113–143 (2022). https://doi.org/10.1007/s10474-022-01273-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-022-01273-8

Key words and phrases

Mathematics Subject Classification

Search

Navigation