The Michigan Mathematical Journal

Mixed Weak Estimates of Sawyer Type for Commutators of Generalized Singular Integrals and Related Operators

Fabio Berra, Marilina Carena, and Gladis Pradolini

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Abstract

We study mixed weak-type inequalities for the commutator [b,T], where b is a BMO function, and T is a Calderón–Zygmund operator. More precisely, we prove that, for every t>0,

uv({xRn:|[b,T](fv)(x)v(x)|>t})CRnΦ(|f(x)|t)u(x)v(x)dx, where Φ(t)=t(1+log+t), uA1, and vA(u). Our technique involves the classical Calderón–Zygmund decomposition, which allows us to give a direct proof without taking into account the associated maximal operator. We use this result to prove an analogous inequality for higher-order commutators.

For a given Young function ϕ we also consider singular integral operators T whose kernels satisfy a Lϕ-Hörmander property, and we find sufficient conditions on ϕ such that a mixed weak estimate holds for T and also for its higher order commutators Tbm.

We also obtain a mixed estimation for a wide class of maximal operators associated to certain Young functions of LlogL type which are in intimate relation with the commutators. This last estimate involves an arbitrary weight u and a radial function v which is not even locally integrable.

Article information

Source
Michigan Math. J., Volume 68, Issue 3 (2019), 527-564.

Dates
Received: 28 July 2017
Revised: 23 February 2018
First available in Project Euclid: 7 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1559894545

Digital Object Identifier
doi:10.1307/mmj/1559894545

Mathematical Reviews number (MathSciNet)
MR3990170

Subjects
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory

Citation

Berra, Fabio; Carena, Marilina; Pradolini, Gladis. Mixed Weak Estimates of Sawyer Type for Commutators of Generalized Singular Integrals and Related Operators. Michigan Math. J. 68 (2019), no. 3, 527--564. doi:10.1307/mmj/1559894545. https://projecteuclid.org/euclid.mmj/1559894545


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References

  • [1] A. Bernardis, E. Dalmasso, and G. Pradolini, Generalized maximal functions and related operators on weighted Musielak–Orlicz spaces, Ann. Acad. Sci. Fenn. Math. 39 (2014), no. 1, 23–50.
  • [2] A. Bernardis, S. Hartzstein, and G. Pradolini, Weighted inequalities for commutators of fractional integrals on spaces of homogeneous type, J. Math. Anal. Appl. 322 (2006), no. 2, 825–846.
  • [3] M. Bramanti and M. C. Cerutti, Commutators of singular integrals and fractional integrals on homogeneous spaces, Harmonic analysis and operator theory (Caracas, 1994), Contemp. Math., 189, pp. 81–94, American Mathematical Society, Providence, RI, 1995.
  • [4] M. Bramanti, M. C. Cerutti, and M. Manfredini, $L^{p}$ estimates for some ultraparabolic operators with discontinuous coefficients, J. Math. Anal. Appl. 200 (1996), no. 2, 332–354.
  • [5] F. Chiarenza, M. Frasca, and P. Longo, Interior $W^{2,p}$ estimates for nondivergence elliptic equations with discontinuous coefficients, Ric. Mat. 40 (1991), no. 1, 149–168.
  • [6] F. Chiarenza, M. Frasca, and P. Longo, $W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 336 (1993), no. 2, 841–853.
  • [7] D. Cruz-Uribe, J. M. Martell, and C. Pérez, Weighted weak-type inequalities and a conjecture of Sawyer, Int. Math. Res. Not. 30 (2005), 1849–1871.
  • [8] D. V. Cruz-Uribe, J. M. Martell, and C. Pérez, Weights, extrapolation and the theory of Rubio de Francia, Oper. Theory Adv. Appl., 215, Birkhäuser/Springer Basel AG, Basel, 2011.
  • [9] L. Diening and M. Růžička, Calderón–Zygmund operators on generalized Lebesgue spaces ${L}^{p(\cdot)}$ and problems related to fluid dynamics, J. Reine Angew. Math. 563 (2003), 197–220.
  • [10] J. Duoandikoetxea, Fourier analysis, Grad. Stud. Math., 29, American Mathematical Society, Providence, RI, 2001. Translated and revised from the 1995 Spanish original by David Cruz-Uribe.
  • [11] J. García-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holl. Math. Stud., 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104.
  • [12] A. M. Kanashiro, G. Pradolini, and O. Salinas, Weighted modular estimates for a generalized maximal operator on spaces of homogeneous type, Collect. Math. 63 (2012), no. 2, 147–164.
  • [13] M. A. Krasnosel’skiĭ and J. B. Rutickiĭ, Convex functions and Orlicz spaces, P. Noordhoff Ltd., Groningen, 1961. Translated from the first Russian edition by Leo F. Boron.
  • [14] K. Li, S. Ombrosi, and C. Pérez, Proof of an extension of E. Sawyer’s conjecture about weighted mixed weak-type estimates, 2017, arXiv:1703.01530.
  • [15] M. Lorente, J. M. Martell, M. S. Riveros, and A. de la Torre, Generalized Hörmander’s conditions, commutators and weights, J. Math. Anal. Appl. 342 (2008), no. 2, 1399–1425.
  • [16] S. Ombrosi and C. Pérez, Mixed weak type estimates: examples and counterexamples related to a problem of E. Sawyer, Colloq. Math. 145 (2016), no. 2, 259–272.
  • [17] C. Pérez, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal. 128 (1995), no. 1, 163–185.
  • [18] C. Pérez, 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 (1995), no. 1, 135–157.
  • [19] C. Pérez, Sharp estimates for commutators of singular integrals via iterations of the Hardy–Littlewood maximal function, J. Fourier Anal. Appl. 3 (1997), no. 6, 743–756.
  • [20] M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Monogr. Textb. Pure Appl. Math., 146, Marcel Dekker, Inc., New York, 1991.
  • [21] C. Rios, The $L^{p}$ Dirichlet problem and nondivergence harmonic measure, Trans. Amer. Math. Soc. 355 (2003), no. 2, 665–687.
  • [22] E. Sawyer, A weighted weak type inequality for the maximal function, Proc. Amer. Math. Soc. 93 (1985), no. 4, 610–614.
  • [23] E. M. Stein and G. Weiss, Interpolation of operators with change of measures, Trans. Amer. Math. Soc. 87 (1958), 159–172.