Annali di Matematica Pura ed Applicata

, Volume 191, Issue 2, pp 339–362 | Cite as

Global W 2, p estimates for nondivergence elliptic operators with potentials satisfying a reverse Hölder condition

  • M. BramantiEmail author
  • L. Brandolini
  • E. Harboure
  • B. Viviani


In this article, we give some a priori \({L^{p}(\mathbb{R}^{n})}\) estimates for elliptic operators in nondivergence form with VMO coefficients and a potential V satisfying an appropriate reverse Hölder condition, generalizing previous results due to Chiarenza–Frasca–Longo to the scope of Schrödinger-type operators. In particular, our class of potentials includes unbounded functions such as nonnegative polynomials. We apply such a priori estimates to derive some global existence and uniqueness results under some additional assumptions on V.


Schrödinger operator Global existence and uniqueness Reverse Hölder condition VMO coefficients Global Lp estimates 

Mathematics Subject Classification (2000)

Primary 35J10 Secondary 35B45 35A05 42B35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bramanti, M.: Commutators of integral operators with positive kernels. Matematiche (Catania) 49 (1994)(1), 149–168 (1995)Google Scholar
  2. 2.
    Chen, Y.-Z., Wu, L.-C.: Second order elliptic equations and elliptic systems. Transl. Math. Monogr. 174, A.M.S. (1991)Google Scholar
  3. 3.
    Chiarenza, F., Frasca, M., Longo, P.: Interior W 2,p-estimates for nondivergence elliptic equations with discontinuous coefficients. Ricerche di Mat. XL, 149–168 (1991)Google Scholar
  4. 4.
    Chiarenza F., Frasca M., Longo P.: W 2,p-solvability of the Dirichlet problem for non divergence elliptic equations with VMO coefficients. Trans. Am. Math. Soc. 336(1), 841–853 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Dziubanski J.: Note on H 1 spaces related to degenerate Schrödinger operators. Ill. J. Math. 49(4), 1271–1297 (2005)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Gehring F.W.: The L p-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130, 265–277 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer Series “Classics in Mathematics” (1976)Google Scholar
  8. 8.
    Guo Z., Li P., Peng L.: L p boundedness of commutators of Riesz transforms associated to Schrödinger operator. J. Math. Anal. Appl. 341, 421–432 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Sobolev Spaces. Graduate Studies in Mathematics, A.M.S. (2008)Google Scholar
  10. 10.
    Kurtz D.S., Wheeden R.L.: Results on weighted norm inequalities for multipliers. Trans. Am. Math. Soc. 255, 343–362 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Lorente M., Martell J.M., Riveros M.S., de la Torre A.: Generalized Hörmander’s conditions, commutators and weights. J. Math. Anal. Appl. 342, 1399–1425 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Martell J.M., Pérez C., Trujillo-González R.: Lack of natural weighted estimates for some singular integral operators. Trans. Amer. Math. Soc. 357(1), 385–396 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Rubio de Francia J.L., Ruiz F.J., Torrea J.L.: Calderón-Zygmund theory for vector-valued functions. Adv. Math. 62, 7–48 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Sarason D.: Functions of vanishing mean oscillation. Trans. Amer. Math. Soc. 207, 391–405 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Segovia C., Torrea J.L.: Vector-valued commutators and applications. Indiana Univ. Math. J. 38(4), 959–971 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Shen Z.: L p estimates for Schrodinger’s operators with certain potentials. Ann. Inst. Fourier t. 45(2), 513–546 (1995)zbMATHCrossRefGoogle Scholar
  17. 17.
    Shen Z.: On the Neumann problem for Schrödinger operators in Lipschitz domains. Indiana Univ. Math. J. 43(1), 143–176 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Stein E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton Univ. Press, Princeton (1993)zbMATHGoogle Scholar
  19. 19.
    Thangavelu S.: Riesz transforms and the wave equation for the Hermite operator. Comm. Partial Differ. Equ. 15(8), 1199–1215 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Vitanza, C.: A new contribution to the W 2,p regularity for a class of elliptic second order equations with discontinuous coefficients. Le Matematiche (Catania) 48(1993)(2), 287–296 (1994)Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag 2011

Authors and Affiliations

  • M. Bramanti
    • 1
    Email author
  • L. Brandolini
    • 2
  • E. Harboure
    • 3
  • B. Viviani
    • 3
  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanItaly
  2. 2.Dipartimento di Ingegneria dell’Informazione e Metodi MatematiciUniversità degli Studi di BergamoDalmineItaly
  3. 3.Instituto de Matemática Aplicada del LitoralCONICET-Universidad Nacional del LitoralSanta FeRepublic of Argentina

Personalised recommendations