Abstract
The purpose of this chapter is to study the properties of the linear extremal function, Λ E (z), which is the upper envelope of plurisubharmonic functions in ℂ n that grow like |z| + o(|z|) and are bounded by 0 on E ⊂ ℝn. The function Λ E (z) is an analogue of the well-known extremal plurisubharmonic function of logarithmic growth obtained when |z| is replaced by logz in the definition. It arises in the study of Phragmén–Lindelöf conditions on algebraic varieties, and the interest is how the growth of Λ E (z) depends on the geometry of the set E. We prove that the linear extremal function can have a linear bound (Λ E (|z|) ≤ Az + B), or a nonlinear bound (e.g., Λ E (z) ≤ Az 3 ∕ 2 + B), or it can be unbounded (Λ E (z) ≡ + ∞). Examples of all three cases are provided. When E is a two-sided cone in ℝn, an exact formula for Λ E (z) is given.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bainbridge, David, Phragmén-Lindelöf estimates for plurisubharmonic functions of linear growth, Thesis, University of Michigan, 1998
Braun, Rüdiger, Meise, Reinhold, Taylor, B. A., A radial Phragmén-Lindelöf Estimate for Plurisubharmonic Functions on Algebraic Varieties, Ann. Polon. Math. 72, 1999, no. 2, 159–179
R. Meise, B. A. Taylor, D. Vogt: Phragmén-Lindelöf principles for algebraic varieties, J. of the Amer. Math. Soc., 11 (1998), 1–39
Meise, R. Taylor, B. A., Vogt, D., Extremal plurisubharmonic functions of linear growth on algebraic varieties, Math. Z., 219, 1995. Vol 4, 515–537
Klimek, Maciej, Pluripotential theory, London Mathematical Society Monographs. New Series, 6, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, 1991
Sibony, Nessim and Wong, Pit Mann, Some results on global analytic sets, Lecture Notes in Math., Séminaire Pierre Lelong, Henri Skoda (Analyse). Années 1978/79, 822, 221–237
Acknowledgements
Portions of this work were part of the author’s doctoral thesis at the University of Michigan written under the direction of B. A. Taylor in 1998 [1].
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Additional information
Dedicated to the memory of Leon Ehrenpreis
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Bainbridge, D. (2013). The Extremal Plurisubharmonic Function for Linear Growth. In: Farkas, H., Gunning, R., Knopp, M., Taylor, B. (eds) From Fourier Analysis and Number Theory to Radon Transforms and Geometry. Developments in Mathematics, vol 28. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4075-8_2
Download citation
DOI: https://doi.org/10.1007/978-1-4614-4075-8_2
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-4074-1
Online ISBN: 978-1-4614-4075-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)