Two Weight Bump Conditions for Matrix Weights

  • David Cruz-Uribe OFS
  • Joshua Isralowitz
  • Kabe Moen


In this paper we extend the theory of two weight, \(A_p\) bump conditions to the setting of matrix weights. We prove two matrix weight inequalities for fractional maximal operators, fractional and singular integrals, sparse operators and averaging operators. As applications we prove quantitative, one weight estimates, in terms of the matrix \(A_p\) constant, for singular integrals, and prove a Poincaré inequality related to those that appear in the study of degenerate elliptic PDEs.


Matrix weights \(A_p\) bump conditions Maximal operators Fractional integral operators Singular integral operators Sparse operators Poincaré inequalities p-Laplacian 

Mathematics Subject Classification

Primary 42B20 42B25 42B35 


  1. 1.
    Berezhnoĭ, E.I.: Two-weighted estimations for the Hardy–Littlewood maximal function in ideal Banach spaces. Proc. Am. Math. Soc. 127(1), 79–87 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bickel, K., Petermichl, S., Wick, B.: Bounds for the Hilbert transform with matrix \(A_2\) weights. J. Funct. Anal. 270(5), 1719–1743 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bickel, K., Wick, B.: A study of the matrix Carleson embedding theorem with applications to sparse operators. J. Math. Anal. Appl. 435(1), 229–243 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Christ, M., Goldberg, M.: Vector \(\text{ A }_{2}\) weights and a Hardy–Littlewood maximal function. Trans. Am. Math. Soc. 353(3), 1995–2002 (2001)CrossRefzbMATHGoogle Scholar
  5. 5.
    Cruz-Uribe, D.: Two weight inequalities for fractional integral operators and commutators. VI International Course of Mathematical Analysis in Andalusia, pp. 25–85 (2016)Google Scholar
  6. 6.
    Cruz-Uribe, D., Martell, J.M., Pérez, C.: Weights, Extrapolation and the Theory of Rubio de Francia, Operator Theory: Advances and Applications, vol. 215. Birkhäuser/Springer Basel AG, Basel (2011)zbMATHGoogle Scholar
  7. 7.
    Cruz-Uribe, D., Moen, K.: A fractional Muckenhoupt–Wheeden theorem and its consequences. Integral Equ. Oper. Theory 76(3), 421–446 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cruz-Uribe, D., Pérez, C.: On the two-weight problem for singular integral operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1(4), 821–849 (2002)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Cruz-Uribe, D., Rodney, S., Rosta, E.: Poincaré inequalities and Neumann problems for the \(p\)-Laplacian, Canad. Math. Bull. To appear (2017). arXiv:1708.03932v1
  10. 10.
    Cruz-Uribe, D., Moen, K., Rodney, S.: Matrix \(\cal{A}_{p}\) weights, degenerate Sobolev spaces, and mappings of finite distortion. J. Geom. Anal. 26(4), 2797–2830 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Culiuc, A., di Plinio, F., Ou, Y.: Uniform sparse domination of singular integrals via dyadic shifts. Math. Res. Lett. To appear (2016). arXiv:1610.01958v2
  12. 12.
    Duoandikoetxea, J.: Fourier Analysis, Graduate Studies in Mathematics, vol. 29. American Mathematical Society, Providence (2001)Google Scholar
  13. 13.
    Duoandikoetxea, J., Martín-Reyes, F.J., Ombrosi, S.: On the \(A_\infty \) conditions for general bases. Math. Z. 282(3–4), 955–972 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, (2001), Reprint of the 1998 editionGoogle Scholar
  15. 15.
    Goldberg, M.: Matrix \(\text{ A }_{p}\) weights via maximal functions. Pac. J. Math. 211(2), 201–220 (2003)CrossRefGoogle Scholar
  16. 16.
    Grafakos, L.: Modern Fourier Analysis, Graduate Texts in Mathematics, vol. 250, 2nd edn. Springer, New York (2008)Google Scholar
  17. 17.
    Hytönen, T., Pérez, C.: Sharp weighted bounds involving \(A_\infty \). Anal. PDE 6(4), 777–818 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hytönen, T., Pérez, C.: The \(L(\log L)^\epsilon \) endpoint estimate for maximal singular integral operators. J. Math. Anal. Appl. 428(1), 605–626 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Isralowitz, J., Kwon, H.K., Pott, S.: Matrix weighted norm inequalities for commutators and paraproducts with matrix symbols. J. Lond. Math. Soc. 96, 243–270 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Isralowitz, J., Moen, K.: Matrix weighted Poincaré inequalities and applications to degenerate elliptic systems, preprint (2016). arXiv:1601.00111
  21. 21.
    Jawerth, B.: Weighted inequalities for maximal operators: linearization, localization and factorization. Am. J. Math. 108(2), 361–414 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lerner, A.K.: On an estimate of Calderón–Zygmund operators by dyadic positive operators. J. Anal. Math. 121(1), 141–161 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Monticelli, D.D., Rodney, S.: Existence and spectral theory for weak solutions of Neumann and Dirichlet problems for linear degenerate elliptic operators with rough coefficients. J. Differ. Equ. 259(8), 4009–4044 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Monticelli, D.D., Rodney, S., Wheeden, R.: Boundedness of weak solutions of degenerate quasilinear equations with rough coefficients. Differ. Integral Equ. 25(1–2), 143–200 (2012)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Monticelli, D.D., Rodney, S., Wheeden, R.: Harnack’s inequality and Hölder continuity for weak solutions of degenerate quasilinear equations with rough coefficients. Nonlinear Anal. 126, 69–114 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Nazarov, F., Petermichl, S., Treil, S., Volberg, A.: Convex body domination and weighted estimates with matrix weights. Adv. Math. 318, 279–306 (2017). arXiv:1701.01907v3
  27. 27.
    Neugebauer, C.J.: Inserting \(A_{p}\)-weights. Proc. Am. Math. Soc. 87(4), 644–648 (1983)zbMATHGoogle Scholar
  28. 28.
    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, 135–157 (1995)CrossRefzbMATHGoogle Scholar
  29. 29.
    Pérez, C.: Two weighted inequalities for potential and fractional type maximal operators. Indiana Univ. Math. J. 43(2), 663–683 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Pérez, C.: Weighted norm inequalities for singular integral operators. J. Lond. Math. Soc. (2) 49(2), 296–308 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Roudenko, S.: Matrix-weighted Besov spaces. Trans. Am. Math. Soc. 355(1), 273–314 (2003). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Sawyer, E.T.: Two weight norm inequalities for certain maximal and integral operators. Harmonic Analysis (Minneapolis, Minn., 1981), pp. 102–127 (1982)Google Scholar
  33. 33.
    Sawyer, E.T., Wheeden, R.: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Am. J. Math. 114(4), 813–874 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Sawyer, E.T., Wheeden, R.: Hölder continuity of weak solutions to subelliptic equations with rough coefficients. Mem. Am. Math. Soc 180(847), x+157 (2006)zbMATHGoogle Scholar
  35. 35.
    Sawyer, E.T., Wheeden, R.: Degenerate Sobolev spaces and regularity of subelliptic equations. Trans. Am. Math. Soc. 362(4), 1869–1906 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • David Cruz-Uribe OFS
    • 1
  • Joshua Isralowitz
    • 2
  • Kabe Moen
    • 1
  1. 1.Department of MathematicsUniversity of AlabamaTuscaloosaUSA
  2. 2.Department of Mathematics and StatisticsSUNY AlbanyAlbanyUSA

Personalised recommendations