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Hardy’s Identities and Inequalities on Cartan-Hadamard Manifolds

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Abstract

We study the Hardy identities and inequalities on Cartan-Hadamard manifolds using the notion of a Bessel pair. These Hardy identities offer significantly more information on the existence/nonexistence of the extremal functions of the Hardy inequalities. These Hardy inequalities are in the spirit of Brezis-Vázquez in the Euclidean spaces. As direct consequences, we establish several Hardy type inequalities that provide substantial improvements as well as simple understandings to many known Hardy inequalities and Hardy-Poincaré-Sobolev type inequalities on hyperbolic spaces in the literature.

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References

  1. Adimurthi, Chaudhuri N., Ramaswamy, M.: An improved Hardy-Sobolev inequality and its application. Proc. Am. Math. Soc. 130(2), 489–505 (2002)

    MathSciNet  MATH  Google Scholar 

  2. Akutagawa, K., Kumura, H.: Geometric relative Hardy inequalities and the discrete spectrum of Schrödinger operators on manifolds. Calc. Var. Part. Differ. Equ. 48, 67–88 (2013)

    MATH  Google Scholar 

  3. Balinsky, A.A., Evans, W.D., Lewis, R.T.: The analysis and geometry of Hardy’s inequality. Universitext, p. 263. Springer, Cham (2015)

    MATH  Google Scholar 

  4. Barbatis, G., Filippas, S., Tertikas, A.: A unified approach to improved \(L^{p}\) Hardy inequalities with best constants. Trans. Am. Math. Soc. 356(6), 2169–2196 (2004)

    MATH  Google Scholar 

  5. Beckner, W.: Pitt’s inequality and the fractional Laplacian: sharp error estimates. Forum Math. 24(1), 177–209 (2012)

    MathSciNet  MATH  Google Scholar 

  6. Berchio, E., D’Ambrosio, L., Ganguly, D., Grillo, G.: Improved \(L^{p}\)-Poincaré inequalities on the hyperbolic space. Nonlinear Anal. 157, 146–166 (2017)

    MathSciNet  MATH  Google Scholar 

  7. Berchio, E., Ganguly, D., Grillo, G.: Sharp Poincaré-Hardy and Poincaré-Rellich inequalities on the hyperbolic space. J. Funct. Anal. 272, 1661–1703 (2017)

    MathSciNet  MATH  Google Scholar 

  8. Berchio, E., Ganguly, D., Grillo, G.: Improved multipolar Poincaré-Hardy inequalities on Cartan-Hadamard manifolds. Ann. Mat. Pura Appl. (4) 199(1), 65–80 (2020)

    MathSciNet  MATH  Google Scholar 

  9. Berchio, E., Ganguly, D., Grillo, G., Pinchover, Y.: An optimal improvement for the Hardy inequality on the hyperbolic space and related manifolds. Proc. Roy. Soc. Edinb. Sect. A 150(4), 1699–1736 (2020)

    MathSciNet  MATH  Google Scholar 

  10. Bosi, R., Dolbeault, J., Esteban, M.J.: Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators. Commun. Pure Appl. Anal. 7(3), 533–562 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Brezis, H., Marcus, M.: Hardy’s inequalities revisited. Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25(1–2), 217–237 (1998)

    Google Scholar 

  12. Brezis, H., Marcus, M., Shafrir, I.: Extremal functions for Hardy’s inequality with weight. J. Funct. Anal. 171(1), 177–191 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Brezis, H., Vázquez, J.L.: Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Complut. Madrid 10(2), 443–469 (1997)

    MathSciNet  MATH  Google Scholar 

  14. Carron, G.: Inégalités de Hardy sur les variétés riemanniennes non-compactes. J. Math. Pures Appl. (9) 76(10), 883–891 (1997)

    MathSciNet  MATH  Google Scholar 

  15. Cazacu, C.: New estimates for the Hardy constants of multipolar Schrödinger operators. Commun. Contemp. Math. 18(5), 1550093 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Cazacu, C., Zuazua, E.: Improved multipolar Hardy inequalities. In: Studies in phase space analysis with applications to PDEs. Progr. Nonlinear Differential Equations Appl., vol. 84, pp. 35–52. Springer, New York (2013)

    Google Scholar 

  17. Chan, H., Ghoussoub, N., Mazumdar, S., Shakerian, S., de Oliveira Faria, L.F.: Mass and extremals associated with the Hardy-Schrödinger operator on hyperbolic space. Adv. Nonlinear Stud. 18(4), 671–689 (2018)

    MathSciNet  MATH  Google Scholar 

  18. D’Ambrosio, L., Dipierro, S.: Hardy inequalities on Riemannian manifolds and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 31, 449–475 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Davies, E.B.: A review of Hardy inequalities. The Maz’ya anniversary collection, Vol. 2 (Rostock, 1998), vol. 110, pp. 55–67, Oper. Theory Adv. Appl., Birkhäuser, Basel, (1999)

  20. Devyver, B., Fraas, M., Pinchover, Y.: Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon. J. Funct. Anal. 266(7), 4422–4489 (2014)

    MathSciNet  MATH  Google Scholar 

  21. Devyver, B., Pinchover, Y.: Optimal \(L^{p}\) Hardy-type inequalities. Ann. Inst. H. Poincaré Anal. Non. Linéaire. 33(1), 93–118 (2016)

    MathSciNet  MATH  Google Scholar 

  22. Druet, O., Hebey, E.: The \(AB\) program in geometric analysis: sharp Sobolev inequalities and related problems. Mem. Am. Math. Soc. 160(761), 98 (2002)

    MathSciNet  MATH  Google Scholar 

  23. Duy, N., Lam, N., Lu, G.: \(p\)-Bessel pairs, Hardy’s identities and inequalities and Hardy-Sobolev inequalities with monomial weights. J. Geom. Anal. 32(4), 109 (2022)

    MathSciNet  MATH  Google Scholar 

  24. Flynn, J.: Sharp Caffarelli-Kohn-Nirenberg-type inequalities on Carnot groups. Adv. Nonlinear Stud. 20(1), 95–111 (2020)

    MathSciNet  MATH  Google Scholar 

  25. Flynn, J., Lam, N., Lu, G.: Sharp Hardy identities and inequalities on Carnot groups. Adv. Nonlinear Stud. 21(2), 281–302 (2021)

    MathSciNet  MATH  Google Scholar 

  26. Flynn, J., Lam, N., Lu, G. \(L^p\) Hardy identities and inequalities with respect to the distance and mean distance to the boundary. Preprint

  27. Flynn, J., Lam, N., Lu, G.: Hardy-Poincaré-Sobolev type inequalities on hyperbolic spaces and related Riemannian manifolds. J. Funct. Anal. 283(12), 109714 (2022)

  28. Frank, R., Seiringer, R.: Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255(12), 3407–3430 (2008)

    MathSciNet  MATH  Google Scholar 

  29. Gallot, S., Hulin, D., Lafontaine, J.: Riemannian geometry, 3rd edn. Springer, Berlin (2004)

    MATH  Google Scholar 

  30. Gazzola, F., Grunau, H.-C., Mitidieri, E.: Hardy inequalities with optimal constants and remainder terms. Trans. Am. Math. Soc. 356(6), 2149–2168 (2004)

    MathSciNet  MATH  Google Scholar 

  31. Gesztesy, F., Littlejohn, L.: Factorizations and Hardy-Rellich-type inequalities. In: Non-linear partial differential equations, mathematical physics, and stochastic analysis, pp. 207–226, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich (2018)

  32. Ghoussoub, N., Moradifam, A.: Bessel pairs and optimal Hardy and Hardy-Rellich inequalities. Math. Ann. 349(1), 1–57 (2011)

    MathSciNet  MATH  Google Scholar 

  33. Ghoussoub, N., Moradifam, A.: Functional inequalities: new perspectives and new applications, Mathematical Surveys and Monographs, vol. 187. American Mathematical Society, Providence (2013)

    MATH  Google Scholar 

  34. Kombe, I., Ozaydin, M.: Improved Hardy and Rellich inequalities on Riemannian manifolds. Trans. Am. Math. Soc. 361, 6191–6203 (2009)

    MathSciNet  MATH  Google Scholar 

  35. Kombe, I., Ozaydin, M.: Hardy-Poincaré, Rellich and uncertainty principle inequalities on Riemannian manifolds. Trans. Am. Math. Soc. 365, 5035–5050 (2013)

    MATH  Google Scholar 

  36. Kristály, A., Szakál, A.: Interpolation between Brezis-Vázquez and Poincaré inequalities on nonnegatively curved spaces: sharpness and rigidities. J. Differ. Equ. 266(10), 6621–6646 (2019)

    MATH  Google Scholar 

  37. Kufner, A., Maligranda, L., Persson, L.-E.: The Hardy inequality. About its history and some related results. Vydavatelský Servis, Pilsen (2007)

    MATH  Google Scholar 

  38. Kufner, A., Persson, L.-E.: Weighted inequalities of Hardy type, p. 357. World Scientific Publishing Co., Inc., River Edge (2003)

  39. Lam, N., Lu, G., Zhang, L.: Factorizations and Hardy’s type identities and inequalities on upper half spaces. Calc. Var. Partial Differ. Equ. 58(6), 183 (2019)

    MathSciNet  MATH  Google Scholar 

  40. Lam, N., Lu, G., Zhang, L.: Geometric Hardy’s inequalities with general distance functions. J. Funct. Anal. 279(8), 108673 (2020)

    MathSciNet  MATH  Google Scholar 

  41. Lu, G., Yang, Q.: Paneitz operators on hyperbolic spaces and high order Hardy-Sobolev-Maz’ya inequalities on half spaces. Am. J. Math. 141(6), 1777–1816 (2019)

    MathSciNet  MATH  Google Scholar 

  42. Lu, G., Yang, Q.: Green’s functions of Paneitz and GJMS operators on hyperbolic spaces and sharp Hardy-Sobolev-Maz’ya inequalities on half spaces. Adv. Math. 398, 108156 (2022)

  43. Ma, X., Wang, X., Yang, Q.: Hardy-Adams Inequalities on \(\mathbb{H}^{2} \times \mathbb{R}^{n-2}\). Adv. Nonlinear Stud. 21(2), 327–345 (2021)

  44. Maz’ya, V.: Sobolev spaces with applications to elliptic partial differential equations. Second, revised and augmented edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 342, p. 866. Springer, Heidelberg (2011)

    Google Scholar 

  45. Muckenhoupt, B.: Hardy’s inequality with weights. Stud. Math. 44, 31–38 (1972)

    MathSciNet  MATH  Google Scholar 

  46. Ngô, Q.A., Nguyen, V.H.: Sharp constant for Poincaré-type inequalities in the hyperbolic space. Acta Math. Vietnam. 44(3), 781–795 (2019)

    MathSciNet  MATH  Google Scholar 

  47. Nguyen, V.H.: New sharp Hardy and Rellich type inequalities on Cartan-Hadamard manifolds and their improvements. Proc. Roy. Soc. Edinb. Sect. A (2017). https://doi.org/10.1017/prm.2019.37

    Article  Google Scholar 

  48. Opic, B., Kufner, A.: Hardy-type inequalities. Pitman Research Notes in Mathematics Series, vol. 219, p. 333. Longman Scientific & Technical, Harlow (1990)

    MATH  Google Scholar 

  49. Sandeep, K., Tintarev, C.: A subset of Caffarelli-Kohn-Nirenberg inequalities in the hyperbolic space \(\mathbb{H} ^{N}\). Ann. Mat. Pura Appl. (4) 196(6), 2005–2021 (2017)

    MathSciNet  MATH  Google Scholar 

  50. Vázquez, J.L., Zuazua, E.: The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential. J. Funct. Anal. 173(1), 103–153 (2000)

    MathSciNet  MATH  Google Scholar 

  51. Yang, Q., Su, D., Kong, Y.: Hardy inequalities on Riemannian manifolds with negative curvature. Commun. Contemp. Math. 16, 1350043 (2014)

    MathSciNet  MATH  Google Scholar 

  52. Wang, J.: \(L^p\) Hardy’s identities and inequalities for Dunkl operators. Adv. Nonlinear Stud. 22(1), 416–435 (2022)

    MathSciNet  MATH  Google Scholar 

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Funding

Joshua Flynn and Guozhen Lu were partly supported by a grant from the Simons Foundation.

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    Joshua Flynn & Guozhen Lu

  2. School of Science & Environment, Grenfell Campus, Memorial University of Newfoundland, Corner Brook, NL, A2H5G4, Canada

    Nguyen Lam

  3. Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, 400076, India

    Saikat Mazumdar

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  1. Joshua Flynn
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  2. Nguyen Lam
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  4. Saikat Mazumdar
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Correspondence to Joshua Flynn.

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Flynn, J., Lam, N., Lu, G. et al. Hardy’s Identities and Inequalities on Cartan-Hadamard Manifolds. J Geom Anal 33, 27 (2023). https://doi.org/10.1007/s12220-022-01079-8

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