Abstract
Let h be a harmonic function defined on a spherical disk. It is shown that Δk|h|2 is nonnegative for all k ∈ N where Δ is the Laplace-Beltrami operator. This fact is generalized to harmonic functions defined on a disk in a normal homogeneous compact Riemannian manifold, and in particular in a symmetric space of the compact type. This complements a similar property for harmonic functions on Rn discovered by the first two authors and is related to strong convexity of the L2-growth function of harmonic functions.
Original language | American English |
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Pages (from-to) | 1613-1622 |
Number of pages | 10 |
Journal | Proceedings of the American Mathematical Society |
Volume | 150 |
Issue number | 4 |
DOIs | |
State | Published - 2022 |
Bibliographical note
Funding Information:The first and second authors were supported by BSF grant no. 2014108.
Funding Information:
Received by the editors April 27, 2021, and, in revised form, June 20, 2021. 2020 Mathematics Subject Classification. Primary 43A85; Secondary 31C05, 22E30. Key words and phrases. Symmetric spaces, harmonic functions, Laplace powers, frequency function, absolute monotonicity, convexity. The second, third, and fourth authors were supported by ISF grant nos. 753/14 and 681/18.
Publisher Copyright:
© 2022 American Mathematical Society
Keywords
- Absolute monotonicity
- Convexity
- Frequency function
- Harmonic functions
- Laplace powers
- Symmetric spaces