Removability results for subharmonic functions, for harmonic functions and for
holomorphic functions

J. Riihentaus

doi:10.15330/ms.46.2.152-158

We begin with an improvement to Blanchet’s extension result for subharmonic functions. With the aid of this improvement we then give extension results both for harmonic and for holomorphic functions. Our results for holomorphic functions are related to Besicovitch’s and Shiffman’s extension results, at least in some sense.

1. Besicovitch A.S. On sufficient conditions for a function to be analytic, and on behavior of analytic functions in the neighborhood of non-isolated singular point, Proc. London Math. Soc., (2), 32 (1931), 1–9.

2. Blanchet P., On removable singularities of subharmonic and plurisubharmonic functions, Complex Variables, 26 (1995), 311–322.

3. Chirka E.M., Complex analytic sets, Kluwer Academic Publisher, Dordrecht, 1989.

4. Federer H., Geometric measure theory, Springer, Berlin, 1969.

5. Harvey R., Polking J., Removable singularities of solutions of linear partial differential equations, Acta Math., 125 (1970), 39–56.

6. Harvey R., Polking J., Extending analytic objects, Comm. Pure and Appl. Mathematics, 28 (1975), 701–727.

7. Hayman W.K., Kennedy P.B., Subharmonic functions, V.I. Academic Press, London, 1976.

8. Helms L.L., Introduction to potential theory, Wiley-Interscience, New York, 1969.

9. Herve M., Analytic and plurisubharmonic functions in finite and infinite dimensional spaces, Lecture Notes in Mathematics, V.198, Springer, Berlin, 1971.

10. Jarnicki M., Pflug P., Extension of holomorphic functions, Walter de Gruyter, Berlin, 2000.

11. Jarnicki M., Pflug P., Separately analytic functions, European Mathematical Society, Zurich, 2011.

12. Khabibullin B.N., A uniqueness theorem for subharmonic functions of finite order, Mat. Sb., 182 (1991), №6, 811.827; English transl. in Math USSR Sbornik, 73 (1992), №1, 195-210.

13. Khabibullin B.N., Completeness of systems of entire functions in spaces of holomorphic functions, Mat. Zametki, 66 (1994), №4, 603.616; English transl. in Math. Notes, 66 (1999), №4, 495-506.

14. Lelong P., Plurisubharmonic functions and positive differential forms, Gordon and Breach, New York, 1969.

15. Polking J., A survey of removable singularities, Seminar on nonlinear partial differential equations, Berkeley, California, 1983. In: Math. Sci. Res. Inst. Publ (ed. S.S. Chern), 2 (1984), 261-292.

16. Rado T., Subharmonic functions, Springer, Berlin, 1937.

17. Riihentaus J., Removable singularities of analytic functions of several complex variables, Math. Z., 32 (1978) 45-54.

18. Riihentaus J., Removable singularities of analytic and meromorphic functions of several complex variables, Colloquium on Complex Analysis, Joensuu, Finland, 1978, (Complex Analysis, Joensuu 1978). In: Proceedings (eds. Ilpo Laine, Olli Lehto, Tuomas Sorvali), Lecture Notes in Mathematics, 747 (1978), 329-342.

19. Riihentaus J., Subharmonic functions, mean value inequality, boundary behavior, nonintegrability and exceptional sets, Workshop on potential theory and free boundary flows, 2003: Kiev, Ukraine. In: Transactions of the Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev, 1 (2004), 169-191.

20. Riihentaus J., An inequality type condition for quasinearly subharmonic functions and applications, Positivity VII, Leiden, 2013, Zaanen Centennial Conference. In: Ordered Structures and Applications: Positivity VII, Trends in Mathematics, 395-414, Springer International Publishing, 2016.

21. Riihentaus J., Exceptional sets for subharmonic functions, J. Basic & Applied Sciences, 11 (2015), 567-571.

22. Riihentaus J., A removability result for holomorphic functions of several complex variables, J. Basic & Applied Sciences, 12 (2016), 50-52.

23. Rudin W., Real and Complex Analysis, Tata McGraw-Hill, New Delhi, 1979.

24. Shiffman B., On the removal of singularities of analytic sets, Michigan Math. J., 15 (1968), 111-120.

25. Ullrich D.C., Removable sets for harmonic functions, Michigan Math. J., 38 (1991), 467-473.

	Removability results for subharmonic functions, for harmonic functions and for holomorphic functions
Author	J. Riihentaus riihentaus@member.ams.org, juhani.riihentaus@uef.fi University of Oulu, Department of Mathematical Sciences, Finland; University of Eastern Finland, Department of Physics and Mathematics
Abstract	We begin with an improvement to Blanchet’s extension result for subharmonic functions. With the aid of this improvement we then give extension results both for harmonic and for holomorphic functions. Our results for holomorphic functions are related to Besicovitch’s and Shiffman’s extension results, at least in some sense.
Keywords	subharmonic function; harmonic function; holomorphic function; Hausdorff measure; exceptional sets
DOI	doi:10.15330/ms.46.2.152-158
Reference	1. Besicovitch A.S. On sufficient conditions for a function to be analytic, and on behavior of analytic functions in the neighborhood of non-isolated singular point, Proc. London Math. Soc., (2), 32 (1931), 1–9. 2. Blanchet P., On removable singularities of subharmonic and plurisubharmonic functions, Complex Variables, 26 (1995), 311–322. 3. Chirka E.M., Complex analytic sets, Kluwer Academic Publisher, Dordrecht, 1989. 4. Federer H., Geometric measure theory, Springer, Berlin, 1969. 5. Harvey R., Polking J., Removable singularities of solutions of linear partial differential equations, Acta Math., 125 (1970), 39–56. 6. Harvey R., Polking J., Extending analytic objects, Comm. Pure and Appl. Mathematics, 28 (1975), 701–727. 7. Hayman W.K., Kennedy P.B., Subharmonic functions, V.I. Academic Press, London, 1976. 8. Helms L.L., Introduction to potential theory, Wiley-Interscience, New York, 1969. 9. Herve M., Analytic and plurisubharmonic functions in finite and infinite dimensional spaces, Lecture Notes in Mathematics, V.198, Springer, Berlin, 1971. 10. Jarnicki M., Pflug P., Extension of holomorphic functions, Walter de Gruyter, Berlin, 2000. 11. Jarnicki M., Pflug P., Separately analytic functions, European Mathematical Society, Zurich, 2011. 12. Khabibullin B.N., A uniqueness theorem for subharmonic functions of finite order, Mat. Sb., 182 (1991), №6, 811.827; English transl. in Math USSR Sbornik, 73 (1992), №1, 195-210. 13. Khabibullin B.N., Completeness of systems of entire functions in spaces of holomorphic functions, Mat. Zametki, 66 (1994), №4, 603.616; English transl. in Math. Notes, 66 (1999), №4, 495-506. 14. Lelong P., Plurisubharmonic functions and positive differential forms, Gordon and Breach, New York, 1969. 15. Polking J., A survey of removable singularities, Seminar on nonlinear partial differential equations, Berkeley, California, 1983. In: Math. Sci. Res. Inst. Publ (ed. S.S. Chern), 2 (1984), 261-292. 16. Rado T., Subharmonic functions, Springer, Berlin, 1937. 17. Riihentaus J., Removable singularities of analytic functions of several complex variables, Math. Z., 32 (1978) 45-54. 18. Riihentaus J., Removable singularities of analytic and meromorphic functions of several complex variables, Colloquium on Complex Analysis, Joensuu, Finland, 1978, (Complex Analysis, Joensuu 1978). In: Proceedings (eds. Ilpo Laine, Olli Lehto, Tuomas Sorvali), Lecture Notes in Mathematics, 747 (1978), 329-342. 19. Riihentaus J., Subharmonic functions, mean value inequality, boundary behavior, nonintegrability and exceptional sets, Workshop on potential theory and free boundary flows, 2003: Kiev, Ukraine. In: Transactions of the Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev, 1 (2004), 169-191. 20. Riihentaus J., An inequality type condition for quasinearly subharmonic functions and applications, Positivity VII, Leiden, 2013, Zaanen Centennial Conference. In: Ordered Structures and Applications: Positivity VII, Trends in Mathematics, 395-414, Springer International Publishing, 2016. 21. Riihentaus J., Exceptional sets for subharmonic functions, J. Basic & Applied Sciences, 11 (2015), 567-571. 22. Riihentaus J., A removability result for holomorphic functions of several complex variables, J. Basic & Applied Sciences, 12 (2016), 50-52. 23. Rudin W., Real and Complex Analysis, Tata McGraw-Hill, New Delhi, 1979. 24. Shiffman B., On the removal of singularities of analytic sets, Michigan Math. J., 15 (1968), 111-120. 25. Ullrich D.C., Removable sets for harmonic functions, Michigan Math. J., 38 (1991), 467-473.
Pages	152-158
Volume	46
Issue	2
Year	2016
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Removability results for subharmonic functions, for harmonic functions and for holomorphic functions