Harmonic and Trace Inequalities in Lipschitz Domains

  • Soumia Touhami
  • Abdellatif Chaira
  • Delfim F.  M. TorresEmail author


We prove boundary inequalities in arbitrary bounded Lipschitz domains on the trace space of Sobolev spaces. For that, we make use of the trace operator, its Moore–Penrose inverse, and of a special inner product. We show that our trace inequalities are particularly useful to prove harmonic inequalities, which serve as powerful tools to characterize the harmonic functions on Sobolev spaces of non-integer order.


Moore–Penrose equality Trace inequalities Harmonic inequalities Lipschitz domains Trace spaces Harmonic functions Hilbert spaces 

2010 Mathematics Subject Classification

47A30 47J20 



This research is part of the first author’s Ph.D. project, which is carried out at Moulay Ismail University, Meknes. It was essentially finished during a visit of Touhami to the Department of Mathematics of University of Aveiro, Portugal, November 2018. The hospitality of the host institution and the financial support of Moulay Ismail University, Morocco, and CIDMA, Portugal, are here gratefully acknowledged. Torres was partially supported by the Portuguese Foundation for Science and Technology (FCT) through CIDMA, project UID/MAT/04106/2019.


  1. 1.
    R.A. Adams, J.J.F. Fournier, Sobolev Spaces (Elsevier/Academic Press, Amsterdam 2003)zbMATHGoogle Scholar
  2. 2.
    E. Carlen, Trace inequalities and quantum entropy: an introductory course, in Entropy and the Quantum. Contemporary Mathematics, vol. 529 (American Mathematical Society, Providence, RI, 2010), pp. 73–140Google Scholar
  3. 3.
    C.D. Collins, J.L. Taylor, Eigenvalue convergence on perturbed Lipschitz domains for elliptic systems with mixed general decompositions of the boundary, J. Differ. Equ. 265(12), 6187–6209 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    M. Costabel, Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19(3), 613–626 (1988)MathSciNetCrossRefGoogle Scholar
  5. 5.
    X. Chen, R. Jiang, D. Yang, Hardy and Hardy-Sobolev spaces on strongly Lipschitz domains and some applications. Anal. Geom. Metr. Spaces 4, 336–362 (2016)MathSciNetzbMATHGoogle Scholar
  6. 6.
    B. Dacorogna, Introduction to the Calculus of Variations, 3rd edn. (Imperial College Press, London, 2015)zbMATHGoogle Scholar
  7. 7.
    R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Functional and Variational Methods, vol. 2 (Springer, Berlin, 1988)Google Scholar
  8. 8.
    L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, vol. 19 (American Mathematical Society, Providence, 1998)Google Scholar
  9. 9.
    E. Gagliardo, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili. Rend. Sem. Mat. Univ. Padova 27, 284–305 (1957)MathSciNetzbMATHGoogle Scholar
  10. 10.
    V.K. Gupta, P. Sharma, Hypergeometric inequalities for certain unified classes of multivalent harmonic functions. Appl. Appl. Math. 13(1), 315–332 (2018)MathSciNetzbMATHGoogle Scholar
  11. 11.
    M. Hayajneh, S. Hayajneh, F. Kittaneh, On some classical trace inequalities and a new Hilbert-Schmidt norm inequality. Math. Inequal. Appl. 21(4), 1175–1183 (2018)MathSciNetzbMATHGoogle Scholar
  12. 12.
    M. Kohr, W.L. Wendland, Variational approach for the Stokes and Navier–Stokes systems with nonsmooth coefficients in Lipschitz domains on compact Riemannian manifolds. Calc. Var. Partial Differ. Equ. 57(6), 165 (2018)Google Scholar
  13. 13.
    J.-Ph. Labrousse, Inverses généralisés d’opérateurs non bornés. Proc. Amer. Math. Soc. 115(1), 125–129 (1992)MathSciNetzbMATHGoogle Scholar
  14. 14.
    C.R. Loga, An extension theorem for matrix weighted Sobolev spaces on Lipschitz domains. Houston J. Math. 43(4), 1209–1233 (2017)MathSciNetzbMATHGoogle Scholar
  15. 15.
    G. Maze, U. Wagner, A note on the weighted harmonic-geometric-arithmetic means inequalities. Math. Inequal. Appl. 15(1), 15–26 (2012)MathSciNetzbMATHGoogle Scholar
  16. 16.
    W. McLean, Strongly Elliptic Systems and Boundary Integral Equations (Cambridge University Press, Cambridge, 2000)zbMATHGoogle Scholar
  17. 17.
    J. Nečas, Les méthodes directes en théorie des équations elliptiques (Masson et Cie, Éditeurs, Paris, 1967)Google Scholar
  18. 18.
    S.M. Nikol’skiı̆, P.I. Lizorkin, Inequalities for harmonic, spherical and algebraic polynomials. Dokl. Akad. Nauk SSSR 289(3), 541–545 (1986)Google Scholar
  19. 19.
    M. Prats, Sobolev regularity of the Beurling transform on planar domains. Publ. Mat. 61(2), 291–336 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces. Lecture Notes of the Unione Matematica Italiana, vol. 3 (Springer, Berlin, 2007)Google Scholar
  21. 21.
    S. Touhami, A. Chaira, D.F.M. Torres, Functional characterizations of trace spaces in Lipschitz domains. Banach J. Math. Anal. 13, 407–426 (2019)MathSciNetCrossRefGoogle Scholar
  22. 22.
    B. Wei, W. Wang, Some inequalities for general L p-harmonic Blaschke bodies. J. Math. Inequal. 10(1), 63–73 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Soumia Touhami
    • 1
  • Abdellatif Chaira
    • 1
  • Delfim F.  M. Torres
    • 2
    Email author
  1. 1.Moulay Ismail UniversityFaculté des Sciences, Laboratoire de Mathématiques et leures Applications, Equipe EDP et Calcul ScientifiqueMeknesMorocco
  2. 2.Center for Research and Development in Mathematics and Applications (CIDMA), Department of MathematicsUniversity of AveiroAveiroPortugal

Personalised recommendations