Skip to main content

Fractional powers of the Schrödinger operator on weigthed Lipschitz spaces

Abstract

In the setting of the semigroup generated by the Schrödinger operator \(L= -\Delta +V\) with the potential V satisfying an appropriate reverse Hölder condition, we consider some non-local fractional differentiation operators. We study their behaviour on suitable weighted smoothness spaces. Actually, we obtain such continuity results for positive powers of L as well as for the mixed operators \(L^{\alpha /2}V^{\sigma /2}\) and \(L^{-\alpha /2}V^{\sigma /2}\) with \(\sigma >\alpha \), together with their adjoints.

This is a preview of subscription content, access via your institution.

References

  1. Bongioanni, B., Cabral, A., Harboure, E.: Regularity of maximal functions associated to a critical radius function. Rev. Unión Mat. Argentina 60(2), 539–566 (2018)

    MathSciNet  MATH  Google Scholar 

  2. Bongioanni, B., Harboure, E., Quijano, P.: Weighted inequalities for Schrödinger type singular integrals. J. Fourier Anal. Appl. 25(3), 595–632 (2019)

    MathSciNet  Article  Google Scholar 

  3. Bongioanni, B., Harboure, E., Quijano, P.: Behaviour of Schrödinger Riesz transforms over smoothness spaces. arXiv:2008.11217

  4. Bongioanni, B., Harboure, E., Salinas, O.: Weighted inequalities for negative powers of Schrödinger operators. J. Math. Anal. Appl. 348(1), 12–27 (2008)

    MathSciNet  Article  Google Scholar 

  5. Bongioanni, B., Harboure, E., Salinas, O.: Classes of weights related to Schrödinger operators. J. Math. Anal. Appl. 373(2), 563–579 (2011)

    MathSciNet  Article  Google Scholar 

  6. Dziubański, J., Garrigós, G., Martínez, T., Torrea, J.L., Zienkiewicz, J.: \(BMO\) spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality. Math. Z. 249(2), 329–356 (2005)

    MathSciNet  Article  Google Scholar 

  7. Dziubański, J., Zienkiewicz, J.: \({H}^p\) spaces for Schrödinger operators. Fourier Anal. Rel. Top. 56, 45–53 (2002)

    MATH  Google Scholar 

  8. Dziubański, J., Zienkiewicz, J.: \({H}^p\) spaces associated with Schrödinger operator with potential from reverse Hölder classes. Colloq. Math. 98(1), 5–38 (2003)

    MathSciNet  Article  Google Scholar 

  9. Ma, T., Stinga, P.R., Torrea, J.L., Zhang, C.: Regularity properties of Schrödinger operators. J. Math. Anal. Appl. 388(2), 817–837 (2012)

    MathSciNet  Article  Google Scholar 

  10. Ma, T., Stinga, P.R., Torrea, J.L., Zhang, C.: Regularity estimates in Hölder spaces for Schrödinger operators via a \(T1\) theorem. Ann. Mat. Pura Appl. 193(2), 561–589 (2014)

    MathSciNet  Article  Google Scholar 

  11. Shen, Z.: \(L^p\) estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier (Grenoble) 45(2), 513–546 (1995)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to P. Quijano.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bongioanni, B., Harboure, E. & Quijano, P. Fractional powers of the Schrödinger operator on weigthed Lipschitz spaces. Rev Mat Complut 35, 515–543 (2022). https://doi.org/10.1007/s13163-021-00393-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13163-021-00393-z

Keywords

  • Schrödinger operator
  • Weights
  • Regularity spaces

Mathematics Subject Classification

  • Primary 42B20
  • Secondary 35J10