Advertisement

Weighted boundedness of multilinear maximal function using Dirac deltas

  • Abhishek GhoshEmail author
  • Saurabh Shrivastava
  • Kalachand Shuin
Article
  • 4 Downloads

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.

Keywords

Maximal function Weighed inequalities Multilinear operators 

Mathematics Subject Classification

42B25 42B20 

Notes

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.

References

  1. 1.
    Aldaz, J.M.: Remarks on the Hardy–Littlewood maximal function. Proc. R. Soc. Edinb. 128 A, 325–328 (1998)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Carlsson, H.: A new proof of the Hardy–Littlewod maximal theorem. Bull. Lond. Math. Soc. 16, 595–596 (1994)CrossRefGoogle Scholar
  3. 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)Google Scholar
  4. 4.
    de Guzmán, M.: Real Variable Methods in Fourier Analysis, Vol. 46. North-Holland Mathematics Studies, Amsterdam (1981)Google Scholar
  5. 5.
    Grafakos, L., Montgomery-Smith, S.: Best constants for uncentered maximal functions. Bull. Lond. Math. Soc. 29, 60–64 (1997)CrossRefzbMATHGoogle Scholar
  6. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lacey, M.T., Thiele, C.: On Calderón’s conjecture. Ann. Math. (2) 149(2), 475–496 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lerner, A.K., Nazarov, F.: Intuitive dyadic calculus: the basics. Expo. Math. (to appear) arXiv:1508.05639
  10. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Melas, A.D.: The best constant for the centered Hardy–Littlewood maximal inequality. Ann. Math. (2) 157(2), 647–688 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Menarguez, M.T., Soria, F.: Weak type \((1,1)\) inequalities of maximal covolution operators. Rend. Circ. Mat. Palermo 41, 342–352 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Moen, K.: Weighted inequalities for multilinear fractional integral operators. Collect. Math. 60(2), 213–238 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Muckenhoupt, B., Wheeden, R.L.: Weighted norm inequalities for singular and fractional integrals. Trans. Am. Math. Soc. 161, 249–258 (1971)Google Scholar
  16. 16.
    Sawyer, E., Wheeden, R.L.: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Am. J. Math. 114(4), 813–874 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton (1970)Google Scholar
  18. 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)Google Scholar
  19. 19.
    Termini, D., Vitanza, C.: Weighted estimates for the Hardy–Littlewood maximal operator and Dirac deltas. Bull. Lond. Math. Soc. 22, 367–374 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2019

Authors and Affiliations

  • Abhishek Ghosh
    • 1
    Email author
  • Saurabh Shrivastava
    • 2
  • Kalachand Shuin
    • 2
  1. 1.Department of Mathematics and StatisticsIndian Institute of Technology KanpurKanpurIndia
  2. 2.Department of MathematicsIndian Institute of Science Education and ResearchBhopalIndia

Personalised recommendations