Research article

Hardy type identities and inequalities with divergence type operators on smooth metric measure spaces

  • Received: 27 February 2024 Revised: 08 April 2024 Accepted: 29 April 2024 Published: 09 May 2024
  • MSC : 53C21, 26D10

  • We gave the Hardy type identities and inequalities for the divergence type operator $ L_{f, V} $ on smooth metric measure spaces. Additionally, we improved a Rellich type inequality by using the improved Hardy type inequality. Our results improved and included many previously known results as special cases.

    Citation: Pengyan Wang, Jiahao Wang. Hardy type identities and inequalities with divergence type operators on smooth metric measure spaces[J]. AIMS Mathematics, 2024, 9(6): 16354-16375. doi: 10.3934/math.2024792

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  • We gave the Hardy type identities and inequalities for the divergence type operator $ L_{f, V} $ on smooth metric measure spaces. Additionally, we improved a Rellich type inequality by using the improved Hardy type inequality. Our results improved and included many previously known results as special cases.



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    [1] G. Carron, Inégalités de Hardy sur les variétés riemanniennes non-compactes, J. Math. Pures Appl., 76 (1997), 883–891.
    [2] I. Kombe, M. Özaydin, Improved Hardy and Rellich inequalities on Riemannian manifolds, Trans. Amer. Math. Soc., 361 (2009), 6191–6203. https://doi.org/10.1090/S0002-9947-09-04642-X doi: 10.1090/S0002-9947-09-04642-X
    [3] I. Kombe, M. Özaydin, Hardy-Poincaré, Rellich and uncertainty principle inequalities on Riemannian manifolds, Trans. Amer. Math. Soc., 365 (2013), 5035–5050. https://doi.org/10.1090/S0002-9947-2013-05763-7 doi: 10.1090/S0002-9947-2013-05763-7
    [4] C. Xia, Hardy and Rellich type inequalities on complete manifolds, J. Math. Anal. Appl., 409 (2014), 84–90. https://doi.org/10.1016/j.jmaa.2013.06.070 doi: 10.1016/j.jmaa.2013.06.070
    [5] X. Huang, D. Ye, First order Hardy inequalities revisited, Commun. Math. Res., 38 (2022), 535–559. https://doi.org/10.4208/cmr.2021-0085 doi: 10.4208/cmr.2021-0085
    [6] L. D'Ambrosio, S. Dipierro, Hardy inequalities on Riemannian manifolds and applications, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 31 (2014), 449–475. https://doi.org/10.1016/j.anihpc.2013.04.004 doi: 10.1016/j.anihpc.2013.04.004
    [7] E. Berchio, D. Ganguly, P. Roychowdhury, On some strong Poincaré inequalities on Riemannian models and their improvements, J. Math. Anal. Appl., 490 (2020), 124213. https://doi.org/10.1016/j.jmaa.2020.124213 doi: 10.1016/j.jmaa.2020.124213
    [8] W. Dai, Y. Hu, Z. Liu, Sharp reversed Hardy-Littlewood-Sobolev inequality with extension kernel, Studia Math., 2023, 1–38. https://doi.org/10.4064/sm220323-26-1 doi: 10.4064/sm220323-26-1
    [9] E. B. Davies, A review of Hardy inequalities, Oper. Theory Adv. Appl., 110 (1999), 55–67.
    [10] F. Faraci, C. Farkas, A. Kristály, Multipolar Hardy inequalities on Riemannian manifolds, ESAIM Control Optim. Calc. Var., 24 (2018), 551–567. https://doi.org/10.1051/cocv/2017057 doi: 10.1051/cocv/2017057
    [11] Y. Jin, S. Shen, Some $L^{p}$-Hardy and $L^{p}$-Rellich type inequalities with remainder terms, J. Aust. Math. Soc., 113 (2022), 79–98. https://doi.org/10.1017/S1446788721000100 doi: 10.1017/S1446788721000100
    [12] A. Kristaly, Sharp uncertainty principles on Riemannian manifolds: The influence of curvature, J. Math. Pures Appl., 119 (2018), 326–346. https://doi.org/10.1016/j.matpur.2017.09.002 doi: 10.1016/j.matpur.2017.09.002
    [13] I. Kombe, A. Yener, Weighted Hardy and Rellich type inequalities on Riemannian manifolds, Math. Nachr., 289 (2016), 994–1004. https://doi.org/10.1002/mana.201500237 doi: 10.1002/mana.201500237
    [14] E. H. A. Thiam, Weighted Hardy inequality on Riemannian manifolds, Commun. Contemp. Math., 18 (2016), 1550072. https://doi.org/10.1142/S0219199715500728 doi: 10.1142/S0219199715500728
    [15] Q. Yang, D. Su, Y. Kong, Hardy inequalities on Riemannian manifolds with negative curvature, Commun. Contemp. Math., 16 (2014), 1–24. https://doi.org/10.1142/S0219199713500430 doi: 10.1142/S0219199713500430
    [16] S. Yin, A. Abdelhakim, Optimal critical exponent $L^{p}$ inequalities of Hardy type on the sphere via Xiao's method, J. Math. Inequal., 16 (2022), 265–272. dx.doi.org/10.7153/jmi-2022-16-19 doi: 10.7153/jmi-2022-16-19
    [17] G. Huang, B. Ma, M. Zhu, A Reilly type integral formula and its applications, Differ. Geom. Appl., 94 (2024), 102136. https://doi.org/10.1016/j.difgeo.2024.102136 doi: 10.1016/j.difgeo.2024.102136
    [18] G. Huang, M. Zhu, Some geometric inequalities on Riemannian manifolds associated with the generalized modified Ricci curvature, J. Math. Phys., 63 (2022), 111508. https://doi.org/10.1063/5.0116994 doi: 10.1063/5.0116994
    [19] G. Huang, H. Li, Gradient estimates and entropy formulae of porous medium and fast diffusion equations for the Witten Laplacian, Pacific J. Math., 268 (2014), 47–78. http://dx.doi.org/10.2140/pjm.2014.268.47 doi: 10.2140/pjm.2014.268.47
    [20] X. D. Li, Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds, J. Math. Pures Appl., 84 (2005), 1295–1361. https://doi.org/10.1016/j.matpur.2005.04.002 doi: 10.1016/j.matpur.2005.04.002
    [21] P. Wang, F. Zeng, Estimates for the first eigenvalue of diffusion-type operators in weighted manifolds, J. Pseudo-Differ. Oper. Appl., 14 (2023). https://doi.org/10.1007/s11868-023-00554-2 doi: 10.1007/s11868-023-00554-2
    [22] G. Huang, M. Zhu, Some integral inequalities on weighted Riemannian manifolds with boundary, Proc. Amer. Math. Soc., 151 (2023), 4961–4970. https://doi.org/10.1090/proc/16479 doi: 10.1090/proc/16479
    [23] J. Li, C. Xia, An integral formula and its applications on sub-static manifolds, J. Differ. Geom., 113 (2019), 493–518. https://doi.org/10.4310/jdg/1573786972 doi: 10.4310/jdg/1573786972
    [24] F. Du, J. Mao, Hardy and Rellich type inequalities on metric measure spaces, J. Math. Anal. Appl., 429 (2015), 354–365. https://doi.org/10.1016/j.jmaa.2015.04.021 doi: 10.1016/j.jmaa.2015.04.021
    [25] Y. Li, A. Abolarinwa, A. H. Alkhaldi, A. Ali, Some inequalities of Hardy type related to Witten-Laplace operator on smooth metric measure spaces, Mathematics, 10 (2022), 4580. https://doi.org/10.3390/math10234580 doi: 10.3390/math10234580
    [26] A. V. Kolesnikov, E. Milman, Brascamp-Lieb-type inequalities on weighted Riemannian manifolds with boundary, J. Geom. Anal., 27 (2017), 1680–1702. https://doi.org/10.1007/s12220-016-9736-5 doi: 10.1007/s12220-016-9736-5
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