Skip to main content
Springer Nature Link
Log in

A Survey of Hardy Type Inequalities on Homogeneous Groups

  • Conference paper
  • First Online:
Mathematical Analysis, its Applications and Computation (ISAAC 2019)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 385))

Abstract

In this review paper, we survey Hardy type inequalities from the point of view of Folland and Stein’s homogeneous groups. Particular attention is paid to Hardy type inequalities on stratified groups which give a special class of homogeneous groups. In this environment, the theory of Hardy type inequalities becomes intricately intertwined with the properties of sub-Laplacians and more general subelliptic partial differential equations. Particularly, we discuss the Badiale-Tarantello conjecture and a conjecture on the geometric Hardy inequality in a half-space of the Heisenberg group with a sharp constant.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
eBook
USD 119.00
Price excludes VAT (USA)
Softcover Book
USD 159.99
Price excludes VAT (USA)
Hardcover Book
USD 159.99
Price excludes VAT (USA)

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    In the case of the Heisenberg group, \(\mathscr {L}\)-gauge is called a Kaplan distance.

References

  1. Adimurthi, A. Sekar, Role of the fundamental solution in Hardy-Sobolev-type inequalities. Proc. R. Soc. Edinb. Sect. A 136(6), 1111–1130 (2006)

    Article  MATH  Google Scholar 

  2. N. Badiale, G. Tarantello, A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics. Arch. Ration. Mech. Anal. 163, 259–293 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. H. Bahouri, J.-Y. Chemin, I. Gallagher, Refined Hardy inequalities. Sup. Pisa Cl. Sci 5, 375–391 (2006)

    MathSciNet  MATH  Google Scholar 

  4. H. Bahouri, J.Y. Chemin, C.J. Xu, Trace and trace lifting theorems in weighted Sobolev spaces. J. Inst. Math. Jussieu 4(4), 509–552 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. A.A. Balinsky, W.D. Evans, R.T. Lewis, The Analysis and Geometry of Hardy’s Inequality. Universitext (Springer, Cham, 2015)

    Book  MATH  Google Scholar 

  6. L. Capogna, D. Danielli, S.D. Pauls, J.T. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem. Progress in Mathematics, vol. 314 (Birkhäuser, Basel, 2007)

    Google Scholar 

  7. L. D’Ambrosio, Some Hardy inequalities on the Heisenberg group. Differ. Equ. 40(4), 552–564 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. L. D’Ambrosio, Hardy-type inequalities related to degenerate elliptic differential operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4(3), 451–486 (2005)

    Google Scholar 

  9. L. Chen, G. Lu, C. Tao. Reverse Stein-Weiss inequalities on the upper half space and the existence of their extremals. Adv. Nonlinear Stud. 19(3), 475–494 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. D. Danielli, N. Garofalo, N. C. Phuc, Hardy-Sobolev type inequalities with sharp constants in Carnot-Carathéodory spaces. Potential Anal. 34, 223–242 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. E.B. Davies, The Maz’ya anniversary Collection. OT Adv. Appl. 55–67 (1999)

    Google Scholar 

  12. J.L. Dyer, A nilpotent Lie algebra with nilpotent automorphism group. Bull. Am. Math. Soc. 76, 52–56 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Filippas, V. Maz’ya, A. Tertikas, Critical Hardy-Sobolev inequalities. J. Math. Pures Appl. 87, 37–56 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. V. Fischer, M. Ruzhansky, Quantization on Nilpotent Lie Groups. Progress in Mathematics, vol. 314 (Birkhäuser, Basel, 2016)

    Google Scholar 

  15. G. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups. Arkiv Mat. 13(2), 161–207 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  16. G. Folland, The Heisenberg group and its relatives in the work of Elias M. Stein. J. Geom. Anal. 13(2), 161–207 (2019)

    Google Scholar 

  17. G.B. Folland, E.M. Stein, Estimates for the \(\overline {\partial _{b}}\) complex and analysis on the Heisenberg group. Commun. Pure Appl. Math. 27, 429–522 (1974)

    Google Scholar 

  18. G.B. Folland, E.M. Stein, Hardy Spaces on Homogeneous Groups. Mathematical Notes, vol. 28 (Princeton University Press, Princeton; University of Tokyo Press, Tokyo, 1982)

    Google Scholar 

  19. V. Franceschi, D. Prandi, Hardy-type inequalities for the Carnot-Carathéodory distance in the Heisenberg group. J. Geom. Anal. (2020). https://doi.org/10.1007/s12220-020-00360-y. arXiv:1903.08486 (2019)

  20. R.L. Frank, M. Loss, The Hardy-Sobolev-Maz’ya inequalities for arbitrary domains. J. Math. Pures Appl. 97, 39–54 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. R.L. Frank, E.H Lieb, Sharp constants in several inequalities on the Heisenberg group. Ann. Math. 176, 349–381 (2012)

    Google Scholar 

  22. N. Garofalo, Geometric second derivative estimates in Carnot groups and convexity. Manuscr. Math. 126, 353–373 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. N. Garofalo, E. Lanconelli, Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation. Ann. Inst. Fourier (Grenoble) 40, 313–356 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  24. J.A. Goldstein, I. Kombe, The Hardy inequality and non-linear parabolic equations on Carnot groups. Nonlinear Anal. 69(12), 4643–4653 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. G.H. Hardy, Notes on some points in the integral calculus (51). Messenger Math. 48, 107–112 (1918)

    Google Scholar 

  26. L. Hörmander, Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  27. Y. Jin, S. Shen, Weighted Hardy and Rellich inequality on Carnot groups. Arch. Math. (Basel) 96(3), 263–271 (2011)

    Google Scholar 

  28. V. Guliyev, R. Mustafayev, A. Serbetci, Stein-Weiss inequalities for the fractional integral operators in Carnot groups and applications. Complex Var. Elliptic Equ. 55(8–10), 847–863 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Kassymov, M. Ruzhansky, D. Suragan, Hardy-Littlewood-Sobolev and Stein-Weiss inequalities on homogeneous Lie groups. Integr. Transforms Special Funct. 30, 643–655 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  30. A. Kassymov, M. Ruzhansky, D. Suragan, Reverse Stein-Weiss, Hardy-Littlewood-Sobolev, Hardy, Sobolev and Caffarelli-Kohn-Nirenberg inequalities on homogeneous groups (2019). arXiv:1912.01460

    Google Scholar 

  31. S. Larson, Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domain in the Heisenberg group. Bull. Math. Sci. 6, 335–352 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. J. Lehrbäck, Hardy inequalities and Assouad dimensions. J. Anal. Math. 131(1), 367–398 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. J. Luan, Q. Yang, A Hardy type inequality in the half-space on Rn and Heisenberg group. J. Math. Anal. Appl. 347, 645–651 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problemes que pose l’hydrodynamique. J. Math. Pures Appl. 12, 1–82 (1933)

    MATH  Google Scholar 

  35. C.C. Martinez, M.A. Sanz, The Theory of Fractional Powers of Operators. North-Holland Mathematics Studies (North-Holland Publishing, Amsterdam, 2001)

    MATH  Google Scholar 

  36. V.G. Maz’ya, Sobolev Spaces (Springer, Berlin, 1985)

    Book  MATH  Google Scholar 

  37. D.T. Nguyen, N. Lam-Hoang, T.A. Nguyen, Hardy and Rellich inequalities with exact missing terms on homogeneous groups. J. Math. Soc. Jpn. 71(4), 1243–1256 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  38. D.T. Nguyen, N. Lam-Hoang, T.A. Nguyen, Hardy-Rellich identities with Bessel pairs. Arch. Math. 113, 95–112 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  39. P. Niu, H. Zhang, Y. Wang, Hardy type and Rellich type inequalities on the Heisenberg group. Proc. Am. Math. Soc. 129(12), 3623–3630 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  40. B. Opic, A. Kufner, Hardy-Type Inequalities Pitman Research Notes, vol. 219 (1990)

    Google Scholar 

  41. T. Ozawa, D. Suragan, Representation formulae for the higher order Steklov and \(L^{2^{m}}\)-Friedrichs inequalities. preprint, 2019

    Google Scholar 

  42. M. Ruzhansky, D. Suragan, Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups. Adv. Math. 317, 799–822 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  43. M. Ruzhansky, D. Suragan, Layer potentials, Kac’s problem, and refined Hardy inequality on homogeneous Carnot groups. Adv. Math. 308, 483–528 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  44. M. Ruzhansky, D. Suragan, On horizontal Hardy, Rellich, Caffarelli-Kohn-Nirenberg and p-sub-Laplacian inequalities on stratified groups. J. Differ. Equ. 262, 1799–1821 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  45. M. Ruzhansky, D. Suragan, N. Yessirkegenov, Sobolev type inequalities, Euler-Hilbert-Sobolev and Sobolev-Lorentz-Zygmund spaces on homogeneous groups. Integr. Equ. Oper. Theory 90, 10 (2018)

    MATH  Google Scholar 

  46. M. Ruzhansky, D. Suragan, N. Yessirkegenov, Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for L p-weighted Hardy inequalities Trans. Am. Math. Soc. Ser. B. 5, 32–62 (2018)

    Article  MATH  Google Scholar 

  47. M. Ruzhansky, B. Sabitbek, D. Suragan, Geometric Hardy and Hardy-Sobolev inequalities on Heisenberg groups. Bull. Math. Sci. 10(3), 2050016 (2020)

    Google Scholar 

  48. M. Ruzhansky, D. Suragan, Anisotropic L2-weighted Hardy and L2-Caffarelli-Kohn-Nirenberg inequalities. Commun. Contemp. Math. 19(6), 1750014 (2017)

    Google Scholar 

  49. M. Ruzhansky, D. Suragan, N. Yessirkegenov, Hardy-Littlewood, Bessel-Riesz, and fractional integral operators in anisotropic Morrey and Campanato spaces. Fract. Calc. Appl. Anal. 21(3), 577–612 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  50. M. Ruzhansky, D. Suragan, Hardy Inequalities on Homogeneous Groups. Progress in Math., vol. 327 (Birkhäuser, Basel, 2019), 588 pp

    Google Scholar 

  51. M. Ruzhansky, N. Yessirkegenov, Hypoelliptic functional inequalities (2018). arXiv:1805.01064v1

    Google Scholar 

  52. E.M. Stein, Some problems in harmonic analysis suggested by symmetric spaces and semi-simple groups. Actes Congres Intern. Math. Nice 1, 179–189 (1970)

    Google Scholar 

  53. S. Secchi, D. Smets, M. Willen, Remarks on a Hardy-Sobolev inequality. C. R. Acad. Sci. Paris Ser. I 336, 811–815 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  54. E.M. Stein, G. Weiss, Fractional integrals on n-dimensional Euclidean Space. J. Math. Mech. 7(4), 503–514 (1958)

    MathSciNet  MATH  Google Scholar 

  55. Q.-H. Yang, Hardy type inequalities related to Carnot-Carathéodory distance on the Heisenberg group. Proc. Am. Math. Soc. 141, 351–362 (2013)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Nazarbayev University, Nur-Sultan City, Republic of Kazakhstan

    Durvudkhan Suragan

Authors
  1. Durvudkhan Suragan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Durvudkhan Suragan .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Aveiro, Aveiro, Portugal

    Paula Cerejeiras

  2. Institute of Applied Analysis, TU Bergakademie Freiberg, Freiberg, Germany

    Michael Reissig

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Suragan, D. (2022). A Survey of Hardy Type Inequalities on Homogeneous Groups. In: Cerejeiras, P., Reissig, M. (eds) Mathematical Analysis, its Applications and Computation. ISAAC 2019. Springer Proceedings in Mathematics & Statistics, vol 385. Springer, Cham. https://doi.org/10.1007/978-3-030-97127-4_4

Download citation

Publish with us

Policies and ethics

Search

Navigation