Calderón-Zygmund estimates for Schrödinger equations revisited
Abstract
We establish a global Calderón-Zygmund estimate for a quasilinear elliptic equation with a potential. If the potential has a reverse Hölder property, then the estimate was known in [6]. In this note, we observe that the estimate remains valid when the potential is merely Lebesgue integrable. Our proof is short and elementary.
Keyword : Calderón-Zygmund estimates, quasilinear Schrödinger equation
How to Cite
Truong, L. X., Trong, N. N., & Do, T. D. (2025). Calderón-Zygmund estimates for Schrödinger equations revisited. Mathematical Modelling and Analysis, 30(2), 224–232. https://doi.org/10.3846/mma.2025.21702
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
B. Bongioanni, E. Harboure and O. Salinas. Commutators of Riesz transforms related to Schrödinger operators. J. Fourier Anal. Appl., 17(1):115–134, 2011. https://doi.org/10.1007/s00041-010-9133-6
M. Bramanti, L. Brandolini, E. Harboure and B. Viviani. Global W2,p estimates for nondivergence elliptic operators with potentials satisfying a reverse Hölder condition. Annali di Matematica, 191(2):339–362, 2012. https://doi.org/10.1007/s10231-011-0186-1
T.D. Do, L.X. Truong and N.N. Trong. Global Hessian estimates in Musielak-Orlicz spaces for a Schrödinger equation. Michigan Math. J., Advance Publication, pp. 1–15, 2024. https://doi.org/10.1307/mmj/20236341
W. Gao and Y. Jiang. Lp estimate for parabolic Schrödinger operator with certain potentials. J. Math. Anal. Appl., 310(1):128–143, 2005. https://doi.org/10.1016/j.jmaa.2005.01.049
M. Lee and J. Ok. Nonlinear Calderón-Zygmund theory involving dual data. Rev. Mat. Iberoamericana, 35(4):1053–1078, 2019. https://doi.org/10.4171/RMI/1078
M. Lee and J. Ok. Interior and boundary W1,q-estimates for quasi-linear elliptic equations of Schrödinger type. J. Differential Equations, 269(5):4406–4439, 2020. https://doi.org/10.1016/j.jde.2020.03.028
G. Pan and L. Tang. Solvability for Schrödinger equations with discontinuous coefficients. J. Funct. Anal., 270(1):88–133, 2016. https://doi.org/10.1016/j.jfa.2015.10.004
Z. Shen. Lp estimates for Schrödinger operators with certain potentials. Annales de l’Institut Fourier, 45(2):513–546, 1995. https://doi.org/10.5802/aif.1463
N.N. Trong, L.X. Truong and T.D. Do. Calderón–Zygmund estimates for a parabolic Schrödinger system on Reifenberg domains. Math. Methods Appl. Sci., 46(4):3797–3820, 2022. https://doi.org/10.1002/mma.8722
N.N. Trong, L.X. Truong and T.D. Do. Higher-order parabolic Schrödinger operators on Lebesgue spaces. Mediterr. J. Math., 19:181, 2022. https://doi.org/10.1007/s00009-022-02082-7
N.N. Trong, L.X. Truong, T.D. Do and T.P.T. Lam. Optimal estimates in Musielak-Orlicz spaces for a parabolic Schrödinger equation. Math. Inequal. Appl., 27(4):909–927, 2024. https://doi.org/10.7153/mia-2024-27-61
M. Bramanti, L. Brandolini, E. Harboure and B. Viviani. Global W2,p estimates for nondivergence elliptic operators with potentials satisfying a reverse Hölder condition. Annali di Matematica, 191(2):339–362, 2012. https://doi.org/10.1007/s10231-011-0186-1
T.D. Do, L.X. Truong and N.N. Trong. Global Hessian estimates in Musielak-Orlicz spaces for a Schrödinger equation. Michigan Math. J., Advance Publication, pp. 1–15, 2024. https://doi.org/10.1307/mmj/20236341
W. Gao and Y. Jiang. Lp estimate for parabolic Schrödinger operator with certain potentials. J. Math. Anal. Appl., 310(1):128–143, 2005. https://doi.org/10.1016/j.jmaa.2005.01.049
M. Lee and J. Ok. Nonlinear Calderón-Zygmund theory involving dual data. Rev. Mat. Iberoamericana, 35(4):1053–1078, 2019. https://doi.org/10.4171/RMI/1078
M. Lee and J. Ok. Interior and boundary W1,q-estimates for quasi-linear elliptic equations of Schrödinger type. J. Differential Equations, 269(5):4406–4439, 2020. https://doi.org/10.1016/j.jde.2020.03.028
G. Pan and L. Tang. Solvability for Schrödinger equations with discontinuous coefficients. J. Funct. Anal., 270(1):88–133, 2016. https://doi.org/10.1016/j.jfa.2015.10.004
Z. Shen. Lp estimates for Schrödinger operators with certain potentials. Annales de l’Institut Fourier, 45(2):513–546, 1995. https://doi.org/10.5802/aif.1463
N.N. Trong, L.X. Truong and T.D. Do. Calderón–Zygmund estimates for a parabolic Schrödinger system on Reifenberg domains. Math. Methods Appl. Sci., 46(4):3797–3820, 2022. https://doi.org/10.1002/mma.8722
N.N. Trong, L.X. Truong and T.D. Do. Higher-order parabolic Schrödinger operators on Lebesgue spaces. Mediterr. J. Math., 19:181, 2022. https://doi.org/10.1007/s00009-022-02082-7
N.N. Trong, L.X. Truong, T.D. Do and T.P.T. Lam. Optimal estimates in Musielak-Orlicz spaces for a parabolic Schrödinger equation. Math. Inequal. Appl., 27(4):909–927, 2024. https://doi.org/10.7153/mia-2024-27-61