Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter July 16, 2020

Two-weighted inequalities for maximal operators related to Schrödinger differential operator

  • Maria Amelia Vignatti , Oscar Salinas EMAIL logo and Silvia Hartzstein
From the journal Forum Mathematicum

Abstract

We introduce classes of pairs of weights closely related to Schrödinger operators, which allow us to get two-weight boundedness results for the Schrödinger fractional integral and its commutators. The techniques applied in the proofs strongly rely on one hand, boundedness results in the setting of finite measure spaces of homogeneous type and, on the other hand, Fefferman–Stein-type inequalities that connect maximal operators naturally associated to Schrödinger operators.

MSC 2010: 42B35; 35J10

Communicated by Christopher D. Sogge


References

[1] D. R. Adams, A note on Riesz potentials, Duke Math. J. 42 (1975), no. 4, 765–778. 10.1215/S0012-7094-75-04265-9Search in Google Scholar

[2] A. Bernardis and O. Salinas, Two-weight norm inequalities for the fractional maximal operator on spaces of homogeneous type, Studia Math. 108 (1994), no. 3, 201–207. 10.4064/sm-108-3-201-207Search in Google Scholar

[3] 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. 10.1016/j.jmaa.2013.04.084Search in Google Scholar

[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. 10.1016/j.jmaa.2010.08.008Search in Google Scholar

[5] B. Bongioanni, E. Harboure and O. Salinas, Commutators of Riesz transforms related to Schrödinger operators, J. Fourier Anal. Appl. 17 (2011), no. 1, 115–134. 10.1007/s00041-010-9133-6Search in Google Scholar

[6] 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. 10.1016/j.jmaa.2012.02.008Search in Google Scholar

[7] S. Chanillo, A note on commutators, Indiana Univ. Math. J. 31 (1982), no. 1, 7–16. 10.1512/iumj.1982.31.31002Search in Google Scholar

[8] D. Cruz-Uribe, New proofs of two-weight norm inequalities for the maximal operator, Georgian Math. J. 7 (2000), no. 1, 33–42. 10.1515/GMJ.2000.33Search in Google Scholar

[9] D. Cruz-Uribe and A. Fiorenza, The A property for Young functions and weighted norm inequalities, Houston J. Math. 28 (2002), no. 1, 169–182. Search in Google Scholar

[10] D. Cruz-Uribe and A. Fiorenza, Endpoint estimates and weighted norm inequalities for commutators of fractional integrals, Publ. Mat. 47 (2003), no. 1, 103–131. 10.5565/PUBLMAT_47103_05Search in Google Scholar

[11] D. Cruz-Uribe and A. Fiorenza, Weighted endpoint estimates for commutators of fractional integrals, Czechoslovak Math. J. 57(132) (2007), no. 1, 153–160. 10.1007/s10587-007-0051-ySearch in Google Scholar

[12] Y. Ding, S. Lu and P. Zhang, Weak estimates for commutators of fractional integral operators, Sci. China Ser. A 44 (2001), no. 7, 877–888. 10.1007/BF02880137Search in Google Scholar

[13] J. Dziubański, G. Garrigós, T. Martínez, J. L. Torrea and J. Zienkiewicz, BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality, Math. Z. 249 (2005), no. 2, 329–356. 10.1007/s00209-004-0701-9Search in Google Scholar

[14] J. Dziubański and J. Zienkiewicz, Hp spaces for Schrödinger operators, Fourier Analysis and Related Topics (Bedlewo 2000), Banach Center Publ. 56, Polish Academy of Sciences, Warsaw (2002), 45–53. 10.4064/bc56-0-4Search in Google Scholar

[15] J. Dziubański and J. Zienkiewicz, Hp spaces associated with Schrödinger operators with potentials from reverse Hölder classes, Colloq. Math. 98 (2003), no. 1, 5–38. 10.4064/cm98-1-2Search in Google Scholar

[16] L. Grafakos, Modern Fourier Analysis, 2nd ed., Grad. Texts in Math. 250, Springer, New York, 2009. 10.1007/978-0-387-09434-2Search in Google Scholar

[17] Z. Guo, P. Li and L. Peng, Lp boundedness of commutators of Riesz transforms associated to Schrödinger operator, J. Math. Anal. Appl. 341 (2008), no. 1, 421–432. 10.1016/j.jmaa.2007.05.024Search in Google Scholar

[18] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, 2nd ed., Cambridge University, Cambridge, 1952. Search in Google Scholar

[19] K. Kurata, An estimate on the heat kernel of magnetic Schrödinger operators and uniformly elliptic operators with non-negative potentials, J. Lond. Math. Soc. (2) 62 (2000), no. 3, 885–903. 10.1112/S002461070000137XSearch in Google Scholar

[20] R. A. Macías and C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. Math. 33 (1979), no. 3, 257–270. 10.1016/0001-8708(79)90012-4Search in Google Scholar

[21] B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261–274. 10.1090/S0002-9947-1974-0340523-6Search in Google Scholar

[22] R. O’Neil, Fractional integration in Orlicz spaces. I, Trans. Amer. Math. Soc. 115 (1965), 300–328. 10.1090/S0002-9947-1965-0194881-0Search in Google Scholar

[23] C. Pérez, Two weighted norm inequalities for Riesz potentials and uniform Lp-weighted Sobolev inequalities, Indiana Univ. Math. J. 39 (1990), no. 1, 31–44. 10.1512/iumj.1990.39.39004Search in Google Scholar

[24] C. Pérez and R. L. Wheeden, Uncertainty principle estimates for vector fields, J. Funct. Anal. 181 (2001), no. 1, 146–188. 10.1006/jfan.2000.3711Search in Google Scholar

[25] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monogr. Textb. Pure Appl. Math. 146, Marcel Dekker, New York, 1991. Search in Google Scholar

[26] E. Sawyer, A two weight weak type inequality for fractional integrals, Trans. Amer. Math. Soc. 281 (1984), no. 1, 339–345. 10.1090/S0002-9947-1984-0719674-6Search in Google Scholar

[27] E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), no. 4, 813–874. 10.2307/2374799Search in Google Scholar

[28] E. T. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308 (1988), no. 2, 533–545. 10.1090/S0002-9947-1988-0930072-6Search in Google Scholar

[29] Z. W. Shen, Lp estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 2, 513–546. 10.5802/aif.1463Search in Google Scholar

[30] L. Tang, Weighted norm inequalities for Schrödinger type operators, Forum Math. 27 (2015), no. 4, 2491–2532. 10.1515/forum-2013-0070Search in Google Scholar

Received: 2019-09-04
Revised: 2020-02-17
Published Online: 2020-07-16
Published in Print: 2020-11-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 18.4.2024 from https://www.degruyter.com/document/doi/10.1515/forum-2019-0243/html
Scroll to top button