Illinois Journal of Mathematics

Two weighted inequalities for operators associated to a critical radius function

B. Bongioanni, E. Harboure, and P. Quijano

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In the general framework of Rd equipped with Lebesgue measure and a critical radius function, we introduce several Hardy–Littlewood type maximal operators and related classes of weights. We prove appropriate two weighted inequalities for such operators as well as a version of Lerner’s inequality for a product of weights. With these tools we are able to prove factored weight inequalities for certain operators associated to the critical radius function. As it is known, the harmonic analysis arising from the Schrödinger operator L=Δ+V, as introduced by Shen, is based on the use of a related critical radius function. When our previous result is applied to this case, it allows to show some inequalities with factored weights for all first and second order Schrödinger–Riesz transforms.

Article information

Illinois J. Math., Volume 64, Number 2 (2020), 227-259.

Received: 21 October 2019
Revised: 15 February 2020
First available in Project Euclid: 1 May 2020

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 35J10: Schrödinger operator [See also 35Pxx]


Bongioanni, B.; Harboure, E.; Quijano, P. Two weighted inequalities for operators associated to a critical radius function. Illinois J. Math. 64 (2020), no. 2, 227--259. doi:10.1215/00192082-8360714.

Export citation


  • [1] B. Bongioanni, A. Cabral, and E. Harboure, Extrapolation for classes of weights related to a family of operators and applications, Potential Anal. 38 (2013), no. 4, 1207–1232.
  • [2] B. Bongioanni, A. Cabral, and E. Harboure, Lerner’s inequality associated to a critical radius function and applications, J. Math. Anal. Appl. 407 2013), no. 1, 35–55.
  • [3] B. Bongioanni, E. Harboure, and P. Quijano, Weighted inequalities for Schrödinger type singular integrals, J. Fourier Anal. Appl. 25 (2019), 595–632.
  • [4] B. Bongioanni, E. Harboure, and O. Salinas, Classes of weights related to Schrödinger operators, J. Math. Anal. Appl. 373 (2011), no. 2, 563–579.
  • [5] B. Bongioanni, E. Harboure, and O. Salinas, Weighted inequalities for commutators of Schrödinger–Riesz transforms, J. Math. Anal. Appl. 392 (2012), no. 1, 6–22.
  • [6] A. Chicco Ruiz and E. Harboure, Weighted norm inequalities for heat-diffusion Laguerre’s semigroups, Math. Z. 257 (2007), no. 2, 329–354.
  • [7] 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, 2011.
  • [8] J. Dziubański and J. Zienkiewicz, Hardy spaces $H^{1}$ associated to Schrödinger operators with potential satisfying reverse Hölder inequality, Rev. Mat. Iberoam. 15 (1999), no. 2, 279–296.
  • [9] C. Pérez, On sufficient conditions for the boundedness of the Hardy–Littlewood maximal operator between weighted $L^{p}$-spaces with different weights, Proc. London Math. Soc. (3) 71 (1995), no. 1, 135–157.
  • [10] Z. Shen, $L^{p}$ estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 2, 513–546.
  • [11] Z. Shen, On fundamental solutions of generalized Schrödinger operators, J. Funct. Anal. 167 (1999), no. 2, 521–564.
  • [12] S. Sugano, $L^{p}$ estimates for some Schrödinger type operators and a Calderón–Zygmund operator of Schrödinger type, Tokyo J. Math. 30 (2007), no. 1, 179–197.