Complex Analysis and Operator Theory

, Volume 13, Issue 2, pp 563–582 | Cite as

The Boundary Behavior of Self-Conjugate Differential Forms in \(C^1\)-domains

  • Francesco SilverioEmail author


This paper deals with the boundary behavior of self-conjugate differential forms. In particular, we state and prove trace theorems and Plemelj-type formulas in \(C^1\)-domains; some remarkable consequences of these results are discussed too. As a further goal, we obtain boundary inequalities extending the classical M. Riesz theorem for conjugate functions.


Self-conjugate differential forms Trace theorems Cauchy integral Plemelj formulas Riesz inequalities 

Mathematics Subject Classification

Primary 42B99 Secondary 31C99 47G10 58A10 



This paper is part of the author’s Ph.D. thesis, written under the supervision of Prof. Alberto Cialdea at University of Basilicata and University of Salento. The author wishes to tank Prof. Alberto Cialdea for valuable comments and suggestions.


  1. 1.
    Cialdea, A.: The brothers Riesz theorem for conjugate differential forms in \({\mathbb{R}}^{n}\). Appl. Anal. 65, 69–94 (1997)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cialdea, A.: On the theory of self-conjugate differential forms. Atti Sem. Mat. Fis. Univ. Modena 66(Suppl.), 595–620 (1998)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Cialdea, A., Silverio, F.: Riesz-type inequalities for conjugate differential forms. J. Math. Anal. Appl. 448(2), 1513–1532 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fichera, G.: Spazi lineari di \(k\)-misure e di forme differenziali. In: Proceedings of International Symposium on Linear Spaces, Jerusalem 1960, Israel Academy of Sciences and Humanities, Pergamon Press, vol. 65, 175–226 (1961)Google Scholar
  5. 5.
    Fichera, G.: On the unification of global and local existence theorems for holomorphic functions of several complex variables. Mem. Acc. Naz. Lincei (8) 18, 61–83 (1986)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Fichera, G., De Vito, L.: Funzioni analitiche di una variabile complessa, 3rd edn. Veschi, Roma (1971)zbMATHGoogle Scholar
  7. 7.
    Folland, G.B.: Introduction to Partial Differential Equations, 2nd edn. Princeton University Press, Princeton (1995)zbMATHGoogle Scholar
  8. 8.
    Hodge, W.V.: A Dirichlet problem for harmonic functionals with applications to analytic varieties. Proc. Lond. Math. Soc. 2–36, 257–303 (1934)CrossRefzbMATHGoogle Scholar
  9. 9.
    Kress, R.: Linear Integral Equations, 3rd edn. Springer, New York (2014)CrossRefzbMATHGoogle Scholar
  10. 10.
    Lovelock, D., Rund, H.: Tensors, Differential Forms, and Variational Principles. Dover, New York (1989)zbMATHGoogle Scholar
  11. 11.
    Malaspina, A.: The Rudin–Carleson theorem for non-homogeneous differential forms. Int. J. Pure Appl. Math. 1(2), 201–212 (2002)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Muskhelishvili, N.I.: Singular Integral Equations. Wolters-Noordhoff Publishing, Groningen (1972)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Sala ConsilinaItaly

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