Local Hessian estimate for second order parabolic equation in Hardy spaces
DOI:
https://doi.org/10.7146/math.scand.a-159735Abstract
We derive an interior estimate up to second orders in Hardy spaces for solutions to the parabolic problem $$ \begin {cases} u_t - \sum _{i, j=1}^n a_{i j} \partial ^2_{ij} u = f & \text {in $\Omega _T$}, \\ u \in h^p(0,T; h^{1,p}(\Omega )) \cap h^{1,p}_{loc }(0,T ; h^{2,p}_{loc }(\Omega )) \end {cases} $$ within an appropriate framework. In the course of proof, we also establish the boundedness results of parabolic singular integrals and their commutators on Hardy spaces which are of independent interest.
References
Bramanti, M., and Cerutti, M. C., $W^1,2_p$ solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients, Comm. Partial Differential Equations 18 (1993), no. 9–10, 1735–1763. https://doi.org/10.1080/03605309308820991
Carbonaro, A., Metafune, G., and Spina, C., Parabolic Schrödinger operators, J. Math. Anal. Appl. 343 (2008), no. 2, 965–974. https://doi.org/10.1016/j.jmaa.2008.02.010
Chiarenza, F., Frasca, M., and Longo, P., Interior $W^2,p$ estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat. 40 (1991), no. 1, 149–168.
Do, T. D., Characterizations of generalized Hardy and BMO spaces via square functions, Kyoto J. Math. 64 (2024), no. 2, 519–555. https://doi.org/10.1215/21562261-2023-0022
Dziubanski, J., Garrigos, G., Martinez, T., Torrea, J. L., and Zienkiewicz, J., BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality, Math. Z. 249 (2005), no. 2, 329–356. https://doi.org/10.1007/s00209-004-0701-9
Fabes, E. R., and Rivière, N., Singular integrals with mixed homogeneity, Studia Math. 27 (1966), 19–38. https://doi.org/10.4064/sm-27-1-19-38
Gao, W., and Jiang, Y., $L^p$ estimate for parabolic Schrödinger operator with certain potentials, J. Math. Anal. Appl. 310 (2005), no. 1, 128–143. https://doi.org/10.1016/j.jmaa.2005.01.049
Gilbarg, D., and Trudinger, N. S., Elliptic partial differential equations of second order, Second edition. Grundlehren der mathematischen Wissenschaften, 224. Springer-Verlag, Berlin, 1983. https://doi.org/10.1007/978-3-642-61798-0
Goldberg, D., A local version of real Hardy spaces, Duke Math. J. 46 (1979), no. 1, 27–42. http://projecteuclid.org/euclid.dmj/1077313253
Ladyzhenskaya, O. A., Solonnikov, V. A., and Uraltseva, N. N., Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, RI, 1968.
Lin, C. C., Convolution operators on Hardy spaces, Studia Math. 120 (1996), no. 1, 53–59. https://doi.org/10.4064/sm-120-1-53-59
Liu, H., Tang, L., and Zhu, H., Weighted Hardy spaces and BMO spaces associated with Schrödinger operators, Math. Nachr. 285 (2012), no. 17–18, 2173–2207. https://doi.org/10.1002/mana.201100323
Liu, Y., Huang, J. Z., and Dong, J., $L^p$ estimates for higher-order parabolic Schrödinger operators with certain nonnegative potentials, Bull. Malays. Math. Sci. Soc. (2) 37 (2014), no. 1, 153–164.
Stein, E. M., Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993.
Sun, Y., and Su, W., Hardy type estimates for commutators of singular integrals, Sci. China Ser. A 48 (2005), no. 4, 563–575. https://doi.org/10.1360/03YS0293
Sun, Y., and Su, W., Interior $h^1$-estimates for second order elliptic equations with vanishing LMO coefficients, J. Funct. Anal. 234 (2006), no. 2, 235–260. https://doi.org/10.1016/j.jfa.2005.10.004
Sun, Y., and Su, W., Solvability in Hardy space for second order elliptic equations, Anal. Theory Appl. 25 (2009), no. 4, 355–368. https://doi.org/10.1007/s10496-009-0355-x
Tang, L., Interior $h^1$ estimates for parabolic equations with LMO coefficients, Canad. J. Math. 62 (2010), no. 1, 202–217. https://doi.org/10.4153/CJM-2010-011-1
Tang, L., and Han, J., $L^p$ boundedness for parabolic Schrödinger type operators with certain nonnegative potentials, Forum Math. 23 (2011), no. 1, 161–179. https://doi.org/10.1515/FORM.2011.007
Trong, N. N., Truong, L. X., and Do, T. D., Boundedness of second-order Riesz transforms on weighted Hardy and BMO spaces associated with Schrödinger operators, C. R. Math. Acad. Sci. Paris 359 (2021), 687–717. https://doi.org/10.5802/crmath.213
Trong, N. N., Truong, L. X., and Do, T. D., Higher-order parabolic Schrödinger operators on Lebesgue spaces, Mediterr. J. Math. 19 (2022), no. 4, Paper No. 181, 22 pp. https://doi.org/10.1007/s00009-022-02082-7
Truong, L. X., Trung, N. D., Trong, N. N., and Do, T. D., Global Hessian estimate for second-order elliptic equation in Hardy spaces, Ric. Mat. 74 (2025), no. 2, 1177–1198. https://doi.org/10.1007/s11587-024-00888-z
