Acknowledgement
This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11871254 and 12071431), the Key Project of Gansu Provincial National Science Foundation (Grant No. 23JRRA1022), the Fundamental Research Funds for the Central Universities (Grant No. lzujbky-2021-ey18) an
References
- P. Auscher, S. C. Hofmann, and J. Martell, Vertical versus conical square functions, Trans. Amer. Math. Soc. 364 (2012), no. 10, 5469-5489. https://doi.org/10.1090/S0002-9947-2012-05668-6
- F. Bernicot and J. Zhao, New abstract Hardy spaces, J. Funct. Anal. 255 (2008), no. 7, 1761-1796. https://doi.org/10.1016/j.jfa.2008.06.018
- R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc. 95 (1960), 263-273. https://doi.org/10.2307/1993291
- K. Bogdan, T. Grzywny, T. Jakubowski, and D. Pilarczyk, Fractional Laplacian with Hardy potential, Comm. Partial Differential Equations 44 (2019), no. 1, 20-50. https://doi.org/10.1080/03605302.2018.1539102
- T. A. Bui and T. Q. Bui, Maximal regularity of parabolic equations associated to generalized Hardy operators in weighted mixed-norm spaces, J. Differential Equations 303 (2021), 547-574. https://doi.org/10.1016/j.jde.2021.09.029
- T. A. Bui and P. D'Ancona, Generalized Hardy operators, Nonlinearity 36 (2023), no. 1, 171-198. https://doi.org/10.1088/1361-6544/ac9c81
- T. A. Bui, P. D'Ancona, X. T. Duong, J. Li, and F. K. Ly, Weighted estimates for powers and smoothing estimates of Schrodinger operators with inverse-square potentials, J. Differential Equations 262 (2017), no. 3, 2771-2807. https://doi.org/10.1016/j.jde.2016.11.008
- T. A. Bui, T. D. Do, and N. N. Trong, Heat kernels of generalized degenerate Schrodinger operators and Hardy spaces, J. Funct. Anal. 280 (2021), no. 1, Paper No. 108785, 49 pp. https://doi.org/10.1016/j.jfa.2020.108785
- T. A. Bui, X. T. Duong, and F. K. Ly, Maximal function characterizations for new local Hardy-type spaces on spaces of homogeneous type, Trans. Amer. Math. Soc. 370 (2018), no. 10, 7229-7292. https://doi.org/10.1090/tran/7289
- T. A. Bui and G. Nader, Hardy spaces associated to generalized Hardy operators and applications, NoDEA Nonlinear Differential Equations Appl. 29 (2022), no. 4, Paper No. 40, 40 pp. https://doi.org/10.1007/s00030-022-00765-4
- L. Chen, J. Martell, and C. Prisuelos-Arribas, Conical square functions for degenerate elliptic operators, Adv. Calc. Var. 13 (2020), no. 1, 75-113. https://doi.org/10.1515/acv-2016-0062
- S. Cho, P. Kim, R. Song, and Z. Vondracek, Factorization and estimates of Dirichlet heat kernels for non-local operators with critical killings, J. Math. Pures Appl. (9) 143 (2020), 208-256. https://doi.org/10.1016/j.matpur.2020.09.006
- D. V. Cruz-Uribe, J. Martell, and C. Perez Moreno, Weights, extrapolation and the theory of Rubio de Francia, Operator Theory: Advances and Applications, 215, Birkhauser/Springer Basel AG, Basel, 2011. https://doi.org/10.1007/978-3-0348-0072-3
- J. Duoandikoetxea, Fourier analysis, translated and revised from the 1995 Spanish original by David Cruz-Uribe, Graduate Studies in Mathematics, 29, Amer. Math. Soc., Providence, RI, 2001. https://doi.org/10.1090/gsm/029
- X. T. Duong and J. Li, Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus, J. Funct. Anal. 264 (2013), no. 6, 1409-1437. https://doi.org/10.1016/j.jfa.2013.01.006
- X. T. Duong and L. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc. 18 (2005), no. 4, 943-973. https://doi.org/10.1090/S0894-0347-05-00496-0
- R. L. Frank, E. H. Lieb, and R. Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schrodinger operators, J. Amer. Math. Soc. 21 (2008), no. 4, 925-950. https://doi.org/10.1090/S0894-0347-07-00582-6
- R. L. Frank, K. Merz, and H. K. H. Siedentop, Equivalence of Sobolev norms involving generalized Hardy operators, Int. Math. Res. Not. IMRN 2021 (2021), no. 3, 2284-2303. https://doi.org/10.1093/imrn/rnz135
- L. Grafakos, Classical Fourier Analysis, second edition, Graduate Texts in Mathematics, 249, Springer, New York, 2008.
- W. Hebisch, Functional calculus for slowly decaying kernels, Preprint, 1995.
- S. C. Hofmann, G. Lu, D. Mitrea, M. Mitrea, and L. Yan, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, Mem. Amer. Math. Soc. 214 (2011), no. 1007, vi+78 pp. https://doi.org/10.1090/S0065-9266-2011-00624-6
- S. C. Hofmann, S. Mayboroda, and A. G. R. McIntosh, Second order elliptic operators with complex bounded measurable coefficients in Lp, Sobolev and Hardy spaces, Ann. Sci. Ec. Norm. Super. (4) 44 (2011), no. 5, 723-800. https://doi.org/10.24033/asens.2154
- T. Jakubowski and J. Wang, Heat kernel estimates of fractional Schrodinger operators with negative Hardy potential, Potential Anal. 53 (2020), no. 3, 997-1024. https://doi.org/10.1007/s11118-019-09795-7
- R. Killip, C. Miao, M. Visan, J. Zhang, and J. Zheng, Sobolev spaces adapted to the Schrodinger operator with inverse-square potential, Math. Z. 288 (2018), no. 3-4, 1273-1298. https://doi.org/10.1007/s00209-017-1934-8
- S. Liang and L. Yan, Riesz transforms associated to Schrodinger operators on weighted Hardy spaces, J. Funct. Anal. 259 (2010), no. 6, 1466-1490. https://doi.org/10.1016/j.jfa.2010.05.015
- S. Liang and L. Yan, A maximal function characterization for Hardy spaces associated to nonnegative self-adjoint operators satisfying Gaussian estimates, Adv. Math. 287 (2016), 463-484. https://doi.org/10.1016/j.aim.2015.09.026
- A. G. R. McIntosh, Operators which have an H∞ functional calculus, in Miniconference on operator theory and partial differential equations (North Ryde, 1986), 210-231, Proc. Centre Math. Anal. Austral. Nat. Univ., 14, Austral. Nat. Univ., Canberra, 1986.
- K. Merz, On scales of Sobolev spaces associated to generalized Hardy operators, Math. Z. 299 (2021), no. 1-2, 101-121. https://doi.org/10.1007/s00209-020-02651-0
- C. Prisuelos-Arribas, Vertical square functions and other operators associated with an elliptic operator, J. Funct. Anal. 277 (2019), no. 12, 108296, 63 pp. https://doi.org/10.1016/j.jfa.2019.108296
- E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Princeton Univ. Press, Princeton, NJ, 1993.
- S. Yang and Z. Yang, A Two-weight boundedness criterion and its applications, arXiv: 2112.04252.