Abstract
The classical Riesz Decomposition Theorem is a powerful tool describing superharmonic functions on compact subsets of \({\mathbb R}^{n}\). There is also the global version of this result dealing with functions superharmonic in \({\mathbb R}^{n}\) and satisfying an additional condition. Recently, a generalization of this result for superbiharmonic functions in \({\mathbb R}^{n}\) was obtained by (J. Anal. Math. 60, 113–133 2006). We consider its further generalization for m-superharmonic functions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Armitage, D.H., Gardiner, S.J.: Classical Potential theory. Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London (2001)
Aronszajn, N., Creese, T.M., Lipkin, L.J.: Polyharmonic functions. Notes taken by Eberhard Gerlach. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1983)
Brauchart, J.S., Dragnev, P.D., Saff, E.B.: Riesz extremal measures on the sphere for axis-supported external fields. J. Math. Anal. Appl. 356(2), 769–792 (2009)
Futamura, T., Kitaura, K., Mizuta, Y.: A Montel type result for super-polyharmonic functions on R N. Potential Anal. 34(1), 89–100 (2011)
Hayman, W.K., Kennedy, P.B.: Subharmonic functions. Vol. I. London Mathematical Society Monographs, No. 9. Academic Press, London-New York (1976)
Hayman, W.K., Korenblum, B.: Representation and uniqueness theorems for polyharmonic functions. J. Anal. Math. 60, 113–133 (1993)
Hörmander, L.: Notions of convexity. Progress in Mathematics, 127. Birkhäuser Boston, Inc., Boston (1994)
Kitaura, K., Mizuta, Y.: Spherical means and Riesz decomposition for superbiharmonic functions. J. Math. Soc. Jpn. 58(2), 521–533 (2006)
Landkof, N.S.: Foundations of modern potential theory. Translated from the Russian by A. P. Doohovskoy. Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer-Verlag, New York-Heidelberg (1972)
Mizuta, Y.: Potential theory in Euclidean spaces. GAKUTO International Series. Mathematical Sciences and Applications, 6. Gakktosho Co., Ltd., Tokyo (1996)
Ransford, T.: Potential theory in the complex plane. London Mathematical Society Student Texts, 28. Cambridge University Press, Cambridge (1995)
Volchkov, V.V.: Integral geometry and convolution equations. Kluwer Academic Publishers, Dordrecht (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tovstolis, A.V. On Riesz Decomposition for Super-Polyharmonic Functions in \({\mathbb R}^{n}\) . Potential Anal 43, 341–360 (2015). https://doi.org/10.1007/s11118-015-9474-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-015-9474-5