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Global Weighted Estimates for Higher Order Schrödinger Operators with Discontinuous Coefficients

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Abstract

We obtain the global weighted estimates for non-divergence and divergence higher order Schrödinger operators with small \(BMO_\theta (\rho )\) (or \(VMO_\theta (\rho )\)) coefficients in the whole space.

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References

  1. Arazy, J., Zelenko, L.: Virtual eigenvalues of the high order Schrödinger operator I. Integral Equ. Oper. Theory. 55, 189–231 (2006)

    Article  Google Scholar 

  2. Arazy, J., Zelenko, L.: Virtual eigenvalues of the high order Schrödinger operator II. Integral Equ. Oper. Theory. 55, 305–345 (2006)

    Article  Google Scholar 

  3. Auscher, P., Ben Ali, B.: Maximal inequalities and Riesz transform estimates on \(L^p\) spaces for Schrödinger operators with nonnegative potentials. Ann. Inst. Fourier (Grenoble) 57, 1975–2013 (2007)

    Article  MathSciNet  Google Scholar 

  4. Badr, N., Ben Ali, B.: \( L^p\) boundedness of the Riesz transform related to Schrödinger operators on a manifold. Ann. Sc. Norm. Super. Pisa Cl. Sci. 8, 725–765 (2009)

    MathSciNet  MATH  Google Scholar 

  5. Barbatis, G., Davies, E.: Sharp bounds on heat kernels of higher uniformly elliptic operators. J. Oper. Theory. 36, 179–198 (1996)

    MathSciNet  MATH  Google Scholar 

  6. Bongioanni, B., Harboure, E., Salinas, O.: Commutators of Riesz transforms related to Schrödinger operators. J. Fourier Anal. Appl. 17, 115–134 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Bramanti, M., Cerutti, M.: \(W_p^{1,2}\) solvability for the Cauchy–Dirichlet problem for parabolic equations with \(VMO\) coefficients. Commun. Part. Differ. Equ. 18, 1735–1763 (1993)

    Article  Google Scholar 

  9. Bramanti, M., Brandolini, L., Harboure, E., Viviani, B.: Global \(W^{2, p}\) estimates for nondivergence elliptic operators with potentials satisfying a reverse Hölder condition. Ann. Mat. Pura Appl. 191, 339–362 (2012)

    Article  MathSciNet  Google Scholar 

  10. Byun, S., Seungjin, R.: Gradient estimates for higher order elliptic equations on nonsmooth domains. J. Differ. Equ. 250, 243–263 (2011)

    Article  MathSciNet  Google Scholar 

  11. Byun, S., Wang, L.: Fourth-order parabolic equations with weak BMO coefficients in Reifenberg domains. J. Differ. Equ. 245, 3217–3252 (2008)

    Article  MathSciNet  Google Scholar 

  12. Chiarenza, F., Frasca, M., Longo, P.: Interior \(W^{2, p}\) estimates for nondivergence form elliptic equations with discontinuous coefficients. Ric. Mat. 40, 149–168 (1991)

    MATH  Google Scholar 

  13. Chiarenza, F., Frasca, M., Longo, P.: \(W^{2, p}\) solvability of the Dirichlet problem for nondivergence elliptic equations with \(VMO\) coefficients. Trans. Am. Math. Soc. 336, 841–853 (1993)

    MATH  Google Scholar 

  14. Cao, J., Liu, Y., Yang, D., Zhang, C.: Gaussian estimates for heat kernels of higher order Schrödinger operators with potentials in generalized schechter classes. arXiv:2012.10888

  15. Dong, H., Kim, D.: \(L_p\) solvalility of divergence type parabolic and elliptic system with partially \(BMO\) coefficients. Calc. Var. 40, 357–389 (2011)

    Article  Google Scholar 

  16. Dong, H., Gallarati, C.: Higher-order elliptic and parabolic equations with VMO assumptions and general boundary conditions. J. Funct. Anal. 274, 1993–2038 (2018)

    Article  MathSciNet  Google Scholar 

  17. Dong, H., Kim, D.: Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains. J. Funct. Anal. 261, 3279–3327 (2011)

    Article  MathSciNet  Google Scholar 

  18. Dong, H., Kim, D.: On the \(L_p\)-solvability of higher order parabolic and elliptic systems with BMO coefficients. Arch. Ration. Mech. Anal. 199, 889–941 (2011)

    Article  MathSciNet  Google Scholar 

  19. Di Fazio, G., Ragusa, M.: Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients. J. Funct. Anal. 112, 241–256 (1993)

    Article  MathSciNet  Google Scholar 

  20. Di Fazio, G., Palagachev, D., Ragusa, M.: Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients. J. Funct. Anal. 166, 179–196 (1999)

    Article  MathSciNet  Google Scholar 

  21. Fan, D., Lu, S., Yang, D.: Regularity in Morrey spaces of strong solutions to nondivergence elliptic equations with \(VMO\) coefficients. Georgian Math. J. 5, 425–440 (1998)

    Article  MathSciNet  Google Scholar 

  22. John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415–426 (1961)

    Article  MathSciNet  Google Scholar 

  23. Kim, D., Krylov, N.: Elliptic differential equations with coefficients measurable with respect to one variable and VMO with respect to the others. SIAM J. Math. Anal. 39, 489–506 (2007)

    Article  MathSciNet  Google Scholar 

  24. Krylov, N.: Parabolic and elliptic equations with VMO coefficients. Commun. Part. Differ. Equ. 32, 453–475 (2007)

    Article  MathSciNet  Google Scholar 

  25. Krylov, N.: Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms. J. Funct. Anal. 250, 521–558 (2007)

    Article  MathSciNet  Google Scholar 

  26. Kurata, K., Sugano, S.: A remark on estimates for uniformly elliptic operators on weighted \(L^p\) spaces and Morrey Spaces. Math Nachr. 209, 137–150 (2000)

    Article  MathSciNet  Google Scholar 

  27. Liu, Y., Dong, F.: Some estimates of higher order Riesz transform related to Schrödinger type operators. Potential Anal. 32, 41–55 (2010)

    Article  MathSciNet  Google Scholar 

  28. Liu, Y., Huang, J.: \( L^p\) estimates for the Schrödinger type operators. Appl. Math. J. Chin. Univ. Ser. B. 26, 412–424 (2011)

    Article  Google Scholar 

  29. Pan, G., Tang, L.: Solvability for Schrödinger equation with discontinuous coefficients. J. Funct. Anal. 270, 88–133 (2016)

    Article  MathSciNet  Google Scholar 

  30. Sarason, D.: Functions of vanishing mean oscillation. Trans. Am. Math. Soc. 207, 391–405 (1975)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  32. Sugano, S.: \(L^p\) estimates for some Schrödinger type operators and a Calderón–Zygmund operator of Schrödinger type. Tokyo J. Math. 30, 179–197 (2007)

    Article  MathSciNet  Google Scholar 

  33. Sugano, S.: Estimates of the fundamental solution for higher order Schrödinger type operators and their applications. J. Funct. Spaces Appl. Art. ID 435480 (2013)

  34. Sugano, S.: A Calderón–Zygmund operator of higher order of Schrödinger type. Math Nachr. 289, 1946–1960 (2016)

    Article  MathSciNet  Google Scholar 

  35. Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  36. Tang, L.: Weighted norm inequalities for Schrödinger type operators. Forum Math. 27, 2491–2532 (2015)

    MathSciNet  MATH  Google Scholar 

  37. Wang, L., Yao, F.: Higher-order nondivergence elliptic and parabolic equations in Sobolev spaces and Orlicz spaces. J. Funct. Anal. 262, 3495–3517 (2012)

    Article  MathSciNet  Google Scholar 

  38. Wang, L., Yao, F.: Global regularity for higher order divergence elliptic and parabolic equations. J. Funct. Anal. 266, 792–813 (2014)

    Article  MathSciNet  Google Scholar 

  39. Zheng, Q., Yao, X.: Higher order Kato class potentials Schrödinger operators. Bull. Lond. Math. Soc. 41, 293–301 (2009)

    Article  MathSciNet  Google Scholar 

  40. Zhong, J.: Harmonic analysis for some Schrödinger type operators, Ph.D. Thesis. Princeton University (1993)

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Acknowledgements

The authors would like to thank the referees for some very valuable suggestions.

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Authors and Affiliations

  1. School of Public Health, Anhui Medical University, Anhui, 230032, People’s Republic of China

    Guixia Pan

  2. LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China

    Lin Tang

  3. Beijing International Studies University, Beijing, 100024, People’s Republic of China

    Hua Zhu

Authors
  1. Guixia Pan
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  2. Lin Tang
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  3. Hua Zhu
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Correspondence to Lin Tang.

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Communicated by Hans G. Feichtinger.

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The research was supported by the NNSF (11771023) and (11801010) of China.

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Pan, G., Tang, L. & Zhu, H. Global Weighted Estimates for Higher Order Schrödinger Operators with Discontinuous Coefficients. J Fourier Anal Appl 27, 85 (2021). https://doi.org/10.1007/s00041-021-09881-8

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  • DOI: https://doi.org/10.1007/s00041-021-09881-8

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