Comptes Rendus
Complex Analysis
The Plemelj–Privalov theorem in Clifford analysis
[Le theorème de Plemelj–Privalov au domaine de l'analyse de Clifford]
Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 223-226.

Cette Note propose une condition géométrique sur une surface de Rn de façon que la transformée de Hilbert sur cette surface, dans le contexte de l'analyse de Clifford, définisse un opérateur borné dans les classes de fonctions de Hölder. Cet résultat généralise le théorème bien connu de Plemelj et Privalov pour des courbes de R2.

This Note gives geometric conditions on a surface of Rn so that the Hilbert transform on that surface in the framework of Clifford analysis defines a bounded operator in the Hölder continuous functions classes. This result provides a generalization of the well-known theorem of Plemelj and Privalov for curves in R2.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.01.029

Ricardo Abreu Blaya 1 ; Juan Bory Reyes 2 ; Tania Moreno García 1

1 Facultad de Informática y Matemática, Universidad de Holguín, Holguín 80100, Cuba
2 Departamento de Matemática, Universidad de Oriente, Santiago de Cuba 90500, Cuba
@article{CRMATH_2009__347_5-6_223_0,
     author = {Ricardo Abreu Blaya and Juan Bory Reyes and Tania Moreno Garc{\'\i}a},
     title = {The {Plemelj{\textendash}Privalov} theorem in {Clifford} analysis},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {223--226},
     publisher = {Elsevier},
     volume = {347},
     number = {5-6},
     year = {2009},
     doi = {10.1016/j.crma.2009.01.029},
     language = {en},
}
TY  - JOUR
AU  - Ricardo Abreu Blaya
AU  - Juan Bory Reyes
AU  - Tania Moreno García
TI  - The Plemelj–Privalov theorem in Clifford analysis
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 223
EP  - 226
VL  - 347
IS  - 5-6
PB  - Elsevier
DO  - 10.1016/j.crma.2009.01.029
LA  - en
ID  - CRMATH_2009__347_5-6_223_0
ER  - 
%0 Journal Article
%A Ricardo Abreu Blaya
%A Juan Bory Reyes
%A Tania Moreno García
%T The Plemelj–Privalov theorem in Clifford analysis
%J Comptes Rendus. Mathématique
%D 2009
%P 223-226
%V 347
%N 5-6
%I Elsevier
%R 10.1016/j.crma.2009.01.029
%G en
%F CRMATH_2009__347_5-6_223_0
Ricardo Abreu Blaya; Juan Bory Reyes; Tania Moreno García. The Plemelj–Privalov theorem in Clifford analysis. Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 223-226. doi : 10.1016/j.crma.2009.01.029. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.01.029/

[1] R. Abreu Blaya; J. Bory Reyes; T. Moreno García Cauchy transform on non-rectifiable surfaces in Clifford analysis, J. Math. Anal. Appl., Volume 339 (2008), pp. 31-44

[2] R. Abreu Blaya; D. Peña Peña; J. Bory Reyes Clifford Cauchy type integrals on Ahlfors–David regular surfaces in Rm+1, Adv. Appl. Clifford Algebras, Volume 13 (2003) no. 2, pp. 133-156

[3] F. Brackx; R. Delanghe; F. Sommen Clifford Analysis, Research Notes in Mathematics, vol. 76, Pitman (Advanced Publishing Program), Boston, 1982

[4] G. David Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Ann. Sci. Ecole Norm. Sup. (4), Volume 17 (1984) no. 1, pp. 157-189 (in French)

[5] G. David; S. Semmes Analysis of and on Uniformly Rectifiable Sets, Mathematical Surveys and Monographs, vol. 38, American Mathematical Society, Providence, RI, 1993

[6] H. Federer Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, vol. 153, Springer-Verlag New York Inc., New York, 1969

[7] E.G. Guseĭnov The Plemelj–Privalov theorem for generalized Hölder classes, Mat. Sb., Volume 183 (1992) no. 2, pp. 21-37 (in Russian); Translation in Russian Acad. Sci. Sb. Math., 75, 1, 1993, pp. 165-182

[8] P. Mattila Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995

[9] V.V. Salaev Direct and inverse estimates for a singular Cauchy integral along a closed curve, Math. Notes, Volume 19 (1976), pp. 221-231

[10] V.V. Salaev; E.G. Guseĭnov; R.K. Seĭfullaev The Plemelj–Privalov theorem, Dokl. Akad. Nauk SSSR, Volume 315 (1990) no. 4, pp. 790-793 (in Russian); Translation in Soviet Math. Dokl., 42, 3, 1991, pp. 849-852

Cité par Sources :

Commentaires - Politique