STRONG CONVEXITY FOR HARMONIC FUNCTIONS ON COMPACT SYMMETRIC SPACES

Gabor Lippner; Dan Mangoubi; Zachary McGuirk; Rachel Yovel

doi:10.1090/proc/15735

STRONG CONVEXITY FOR HARMONIC FUNCTIONS ON COMPACT SYMMETRIC SPACES

Gabor Lippner, Dan Mangoubi, Zachary McGuirk, Rachel Yovel

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let h be a harmonic function defined on a spherical disk. It is shown that Δ^k|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
Pages (from-to)	1613-1622
Number of pages	10
Journal	Proceedings of the American Mathematical Society
Volume	150
Issue number	4
DOIs	https://doi.org/10.1090/proc/15735
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

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10.1090/proc/15735

Cite this

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title = "STRONG CONVEXITY FOR HARMONIC FUNCTIONS ON COMPACT SYMMETRIC SPACES",

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.",

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author = "Gabor Lippner and Dan Mangoubi and Zachary McGuirk and Rachel Yovel",

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doi = "10.1090/proc/15735",

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AU - Yovel, Rachel

N1 - 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

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AB - 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.

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KW - Convexity

KW - Frequency function

KW - Harmonic functions

KW - Laplace powers

KW - Symmetric spaces

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STRONG CONVEXITY FOR HARMONIC FUNCTIONS ON COMPACT SYMMETRIC SPACES

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Keywords

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