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Weighted boundedness of multilinear maximal function using Dirac deltas

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Abstract

In this article we extend a method of Miguel de Guzmán involving boundedness properties of maximal functions using Dirac deltas to multilinear setting. This method involves estimating maximal functions over finite linear combination of Dirac deltas. As an application, we obtain end-point weighted boundedness of the multilinear Hardy–Littlewood fractional maximal function with respect to multilinear weights.

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References

  1. Aldaz, J.M.: Remarks on the Hardy–Littlewood maximal function. Proc. R. Soc. Edinb. 128 A, 325–328 (1998)

    MathSciNet  MATH  Google Scholar 

  2. Carlsson, H.: A new proof of the Hardy–Littlewod maximal theorem. Bull. Lond. Math. Soc. 16, 595–596 (1994)

    Article  Google Scholar 

  3. Duoandikoetxea, J.: Fourier Analysis. Translated and revised from the 1995 Spanish original by David Cruz–Uribe. Graduate Studies in Mathematics. 29 American Mathematical Society, Providence (2001)

  4. de Guzmán, M.: Real Variable Methods in Fourier Analysis, Vol. 46. North-Holland Mathematics Studies, Amsterdam (1981)

  5. Grafakos, L., Montgomery-Smith, S.: Best constants for uncentered maximal functions. Bull. Lond. Math. Soc. 29, 60–64 (1997)

    Article  Google Scholar 

  6. Lacey, M.T.: The bilinear maximal functions map into \(L^p\) for \(2/3<p\le 1\). Ann. Math. (2) 151(1), 35–57 (2000)

    Article  MathSciNet  Google Scholar 

  7. Lacey, M.T., Thiele, C.: \(L^p\) estimates on the bilinear Hilbert transform for \(2<p<\infty \). Ann. Math. (2) 146(3), 693–724 (1997)

    Article  MathSciNet  Google Scholar 

  8. Lacey, M.T., Thiele, C.: On Calderón’s conjecture. Ann. Math. (2) 149(2), 475–496 (1999)

    Article  MathSciNet  Google Scholar 

  9. Lerner, A.K., Nazarov, F.: Intuitive dyadic calculus: the basics. Expo. Math. (to appear) arXiv:1508.05639

  10. Lerner, A.K., Ombrosi, S., Perez, C., Torres, R.H., Trujillo-Gonzalez, R.: New maximal functions and multiple weights for the multilinear Calderon–Zygmund theory. Adv. Math. 220, 1222–1264 (2009)

    Article  MathSciNet  Google Scholar 

  11. Melas, A.D.: The best constant for the centered Hardy–Littlewood maximal inequality. Ann. Math. (2) 157(2), 647–688 (2003)

    Article  MathSciNet  Google Scholar 

  12. Menarguez, M.T., Soria, F.: Weak type \((1,1)\) inequalities of maximal covolution operators. Rend. Circ. Mat. Palermo 41, 342–352 (1992)

    Article  MathSciNet  Google Scholar 

  13. Moen, K.: Weighted inequalities for multilinear fractional integral operators. Collect. Math. 60(2), 213–238 (2009)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. Muckenhoupt, B., Wheeden, R.L.: Weighted norm inequalities for singular and fractional integrals. Trans. Am. Math. Soc. 161, 249–258 (1971)

    Article  MathSciNet  Google Scholar 

  16. Sawyer, E., Wheeden, R.L.: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Am. J. Math. 114(4), 813–874 (1992)

    Article  MathSciNet  Google Scholar 

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

  18. Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, With the assistance of Timothy S. Murphy. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton (1993)

  19. Termini, D., Vitanza, C.: Weighted estimates for the Hardy–Littlewood maximal operator and Dirac deltas. Bull. Lond. Math. Soc. 22, 367–374 (1990)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We thank the anonymous referee for his/her valuable suggestions that helped to improve the article. Also, the first author expresses his sincere gratitude to his thesis supervisor Prof. Parasar Mohanty for many fruitful discussions. Funding was provided by Ministry of Human Resource Development (Grant No. MHRD Gate-2013). The third author thanks Council of Scientific and Industrial Research for their financial support.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016, India

    Abhishek Ghosh

  2. Department of Mathematics, Indian Institute of Science Education and Research, Bhopal, 462066, India

    Saurabh Shrivastava & Kalachand Shuin

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  1. Abhishek Ghosh
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  2. Saurabh Shrivastava
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  3. Kalachand Shuin
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Correspondence to Abhishek Ghosh.

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Ghosh, A., Shrivastava, S. & Shuin, K. Weighted boundedness of multilinear maximal function using Dirac deltas. Rend. Circ. Mat. Palermo, II. Ser 69, 273–285 (2020). https://doi.org/10.1007/s12215-019-00401-8

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