Polyharmonic functions on trees

Cohen, Julie; Colonna, Flavia; Gowrisankaran, Kohur; Singman, David

doi:10.1353/ajm.2002.0027

Cited by 20 publications

(27 citation statements)

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“…We assume that , where is the -spectral radius of P and is its -spectrum. Close to the spirit of Korányi and Picardello [ 11 ], we extend their results from -harmonic to -polyharmonic functions, and results of the abovementioned work [ 5 ] from ordinary polyharmonic functions, i.e. , to general complex in the -resolvent set of P .…”

Section: Introductionmentioning

confidence: 65%

“…A smaller body of work is available on the discrete counterpart, where the Laplacian is a difference operator arising from a reversible Markov chain transition matrix on a graph. Regarding boundary integral representations comparable to ( 1 ), Cohen et al [ 5 ] have provided such a result concerning polyharmonic functions for the simple random walk operator on a homogeneous tree. This has recently been generalised by Picardello and Woess [ 12 ] to arbitrary nearest neighbour transition operators on arbitrary trees which do not need to be locally finite: [ 12 ] provides a boundary integral representation for -polyharmonic functions for suitable complex .…”

Section: Introductionmentioning

confidence: 99%

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Boundary behaviour of $$\lambda $$-polyharmonic functions on regular trees

Sava-Huss

Woess

2020

Annali di Matematica

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This paper studies the boundary behaviour of $$\lambda $$ λ -polyharmonic functions for the simple random walk operator on a regular tree, where $$\lambda $$ λ is complex and $$|\lambda |> \rho $$ | λ | > ρ , the $$\ell ^2$$ ℓ 2 -spectral radius of the random walk. In particular, subject to normalisation by spherical, resp. polyspherical functions, Dirichlet and Riquier problems at infinity are solved, and a non-tangential Fatou theorem is proved.

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Section: Introductionmentioning

confidence: 65%

Section: Introductionmentioning

confidence: 99%

Boundary behaviour of $$\lambda $$-polyharmonic functions on regular trees

Sava-Huss

Woess

2020

Annali di Matematica

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“…In [5], we introduced the space B α , for all α 0, for the purpose of studying the boundary behavior of polyharmonic functions on homogeneous trees of degree q + 1. The space B α turns out to be precisely the Besov-Lipschitz space B α 1,1 defined in [4] which can be identified with the space of distributions ν such that v =e…”

Section: The Space Bmentioning

confidence: 99%

Distributions and measures on the boundary of a tree

Cohen

Colonna

Singman

2004

Journal of Mathematical Analysis and Applications

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In this paper, we analyze the space D of distributions on the boundary Ω of a tree and its subspace B 0 , which was introduced in [Amer. J. Math. 124 (2002Math. 124 ( ) 999-1043 in the homogeneous case for the purpose of studying the boundary behavior of polyharmonic functions. We show that if µ ∈ B 0 , then µ is a measure which is absolutely continuous with respect to the natural probability measure λ on Ω, but on the other hand there are measures absolutely continuous with respect to λ which are not in B 0 . We then give the definition of an absolutely summable distribution and prove that a distribution can be extended to a complex measure on the Borel sets of Ω if and only if it is absolutely summable. This is also equivalent to the condition that the distribution have finite total variation. Finally, we show that for a distribution µ, Ω decomposes into two subspaces. On one of them, a union of intervals A µ , µ restricted to any finite union of intervals extends to a complex measure and on A µ we give a version of the Jordan, Hahn, and Lebesgue-Radon-Nikodym decomposition theorems. We also show that there is no interval in the complement of A µ in which any type of decomposition theorem is possible. All the results in this article can be generalized to results on good (in particular, compact infinite) ultrametric spaces, for example, on the p-adic integers and the p-adic rationals. 

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“…The transient theory was largely developed in [24] where Cartier considers basic properties of harmonic and superharmonic functions, integral representation of positive harmonic functions, and limit theorems of positive superharmonic functions at the boundary of the tree along random paths. Limit theorems along deterministic paths are considered in [26], [34] and [35].…”

Section: Introductionmentioning

confidence: 99%

“…For our other collaborations which deal with potential theory on trees but which we do not survey here, see [16], [20], [26], [27], [35].…”

Section: Introductionmentioning

confidence: 99%

Potential theory on trees and mutliplication operators

Singman¹

2012

Complex Analysis and Potential Theory

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The article surveys a number of potential theory results in the discrete setting of trees and in an application to complex analysis. On trees for which the associated random walk is recurrent, we discuss Riesz decomposition, flux, a type of potential called H-potential, and present a new result dealing with the boundary behaviour of H-potentials on a specific recurrent homogeneous tree. On general trees we discuss Brelot structures and their classification. On transient homogeneous trees we discuss clamped and simplysupported biharmonic Green functions.We also describe an application of potential theory, namely a certain minimum principle for multiply superharmonic functions, that is used to prove a result concerning the norm of a class of multiplication operators from H ∞ (D) to the Bloch space on D, where D is a bounded symmetric domain. The proof of the minimum principle involves the use of the Cartan-Brelot topology.2010 Mathematics Subject Classification. Primary 31C20; Secondary 31A30, 31C05, 31D05.

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Polyharmonic functions on trees

Cited by 20 publications

References 19 publications

Boundary behaviour of $$\lambda $$-polyharmonic functions on regular trees

Boundary behaviour of $$\lambda $$-polyharmonic functions on regular trees

Distributions and measures on the boundary of a tree

Potential theory on trees and mutliplication operators

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