scholarly journals Lp–Lrestimates for the Poisson semigroup on homogeneous trees

1998 ◽  
Vol 64 (1) ◽  
pp. 20-32 ◽  
Author(s):  
Alberto G. Setti
Keyword(s):  
Sharp Estimate ◽  
Homogeneous Tree ◽  
Homogeneous Trees ◽  
Poisson Semigroup ◽  
Operator Norms

AbstractLet be a homogeneous tree of degree at least three. In this paper we investigate for which values of p and r the (σθ)-Poisson semigroup is Lp – Lr,-bounded, and we sharp estimate for the corresponding operator norms.

scholarly journals Unitarization of the Horocyclic Radon Transform on Homogeneous Trees

2021 ◽  
Vol 27 (5) ◽  
Author(s):  
Francesca Bartolucci ◽  
Filippo De Mari ◽  
Matteo Monti
Keyword(s):  
Radon Transform ◽  
Homogeneous Tree ◽  
Homogeneous Trees ◽  
Regular Representations ◽  
Group Of Isometries

AbstractFollowing previous work in the continuous setup, we construct the unitarization of the horocyclic Radon transform on a homogeneous tree X and we show that it intertwines the quasi regular representations of the group of isometries of X on the tree itself and on the space of horocycles.

scholarly journals A theorem of Roe and Strichartz on homogeneous trees

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sumit Kumar Rano
Keyword(s):  
Laplace Operator ◽  
Infinite Sequence ◽  
Homogeneous Tree ◽  
Poisson Transform ◽  
Formula One ◽  
Homogeneous Trees ◽  
Suitable Norm ◽  
The Laplace Operator ◽  
Sequence Of Functions

Abstract Let 𝔛 {\mathfrak{X}} be a homogeneous tree and let ℒ {\mathcal{L}} be the Laplace operator on 𝔛 {\mathfrak{X}} . In this paper, we address problems of the following form: Suppose that { f k } k ∈ ℤ {\{f_{k}\}_{k\in\mathbb{Z}}} is a doubly infinite sequence of functions in 𝔛 {\mathfrak{X}} such that for all k ∈ ℤ {k\in\mathbb{Z}} one has ℒ ⁢ f k = A ⁢ f k + 1 {\mathcal{L}f_{k}=Af_{k+1}} and ∥ f k ∥ ≤ M {\lVert f_{k}\rVert\leq M} for some constants A ∈ ℂ {A\in\mathbb{C}} , M > 0 {M>0} and a suitable norm ∥ ⋅ ∥ {\lVert\,\cdot\,\rVert} . From this hypothesis, we try to infer that f 0 {f_{0}} , and hence every f k {f_{k}} , is an eigenfunction of ℒ {\mathcal{L}} . Moreover, we express f 0 {f_{0}} as the Poisson transform of functions defined on the boundary of 𝔛 {\mathfrak{X}} .

scholarly journals SCHUR MULTIPLIERS AND SPHERICAL FUNCTIONS ON HOMOGENEOUS TREES

2010 ◽  
Vol 21 (10) ◽  
pp. 1337-1382 ◽  
Author(s):  
U. HAAGERUP ◽  
T. STEENSTRUP ◽  
R. SZWARC
Keyword(s):  
Fourier Multiplier ◽  
Spherical Functions ◽  
Sufficient Condition ◽  
Homogeneous Tree ◽  
Necessary And Sufficient Condition ◽  
Closed Expression ◽  
Radial Functions ◽  
Schur Multipliers ◽  
Homogeneous Trees ◽  
Necessary And Sufficient

Let X be a homogeneous tree of degree q + 1 (2 ≤ q ≤ ∞) and let ψ : X × X → ℂ be a function for which ψ(x, y) only depends on the distance between x, y ∈ X. Our main result gives a necessary and sufficient condition for such a function to be a Schur multiplier on X × X. Moreover, we find a closed expression for the Schur norm ||ψ||S of ψ. As applications, we obtaina closed expression for the completely bounded Fourier multiplier norm ||⋅||M0A(G) of the radial functions on the free (non-abelian) group 𝔽N on N generators (2 ≤ N ≤ ∞) and of the spherical functions on the q-adic group PGL2(ℚq) for every prime number q.

scholarly journals The range of the Helgason-Fourier transformation on homogeneous trees

1999 ◽  
Vol 59 (2) ◽  
pp. 237-246 ◽  
Author(s):  
Michael Cowling ◽  
Alberto G. Setti
Keyword(s):  
Reference Point ◽  
Fourier Transformation ◽  
Symmetric Spaces ◽  
Closed Ball ◽  
Homogeneous Tree ◽  
Homogeneous Trees ◽  
Fixed Reference ◽  
Decreasing Functions

Let be a homogeneous tree, o be a fixed reference point in , and be the closed ball of radius N in centred at o. In this paper we characterise the image under the Helgason–Fourier transformation ℋ of , the space of functions supported in , and of , the space of rapidly decreasing functions on . In both cases our results are counterparts of known results for the Helgason–Fourier transformation on noncompact symmetric spaces.

scholarly journals Measurable Events Indexed by Trees

2012 ◽  
Vol 21 (3) ◽  
pp. 374-411 ◽  
Author(s):  
PANDELIS DODOS ◽  
VASSILIS KANELLOPOULOS ◽  
KONSTANTINOS TYROS
Keyword(s):  
Finite Subset ◽  
Probability Space ◽  
Homogeneous Tree ◽  
Homogeneous Trees ◽  
Infinite Height ◽  
Probability Spaces ◽  
Image Position

A tree T is said to be homogeneous if it is uniquely rooted and there exists an integer b ≥ 2, called the branching number of T, such that every t ∈ T has exactly b immediate successors. We study the behaviour of measurable events in probability spaces indexed by homogeneous trees.Precisely, we show that for every integer b ≥ 2 and every integer n ≥ 1 there exists an integer q(b,n) with the following property. If T is a homogeneous tree with branching number b and {At:t ∈ T} is a family of measurable events in a probability space (Ω,Σ,μ) satisfying μ(At)≥ϵ>0 for every t ∈ T, then for every 0<θ<ϵ there exists a strong subtree S of T of infinite height, such that for every finite subset F of S of cardinality n ≥ 1 we have In fact, we can take q(b,n)= ((2b−1)2n−1−1)·(2b−2)−1. A finite version of this result is also obtained.

scholarly journals Special series of unitary representations of groups acting on homogeneous trees

1986 ◽  
Vol 40 (1) ◽  
pp. 83-88 ◽  
Author(s):  
Anna Maria Mantero ◽  
Anna Zappa
Keyword(s):  
Discrete Subgroup ◽  
Unitary Representations ◽  
Homogeneous Tree ◽  
Special Series ◽  
Representations Of Groups ◽  
Homogeneous Trees

AbstractLet G be a group acting faithfully on a homogeneous tree of order p + 1, p > 1. Let be the space of functions on the Poission boundary ω, of zero mean on ω. When p is a prime. G is a discrete subgroup of PGL2(Qp) of finite covolume. The representations of the special series of PGL2(Qp), Which are irreducible and unitary in an appropriate completion of , are shown to be reducible when restricted to G. It is proved that these representations of G are algebraically reducible on and topologically irreducible on endowed with the week topology.

scholarly journals Riesz means on homogeneous trees

2021 ◽  
Vol 8 (1) ◽  
pp. 60-65
Author(s):  
Effie Papageorgiou
Keyword(s):  
Homogeneous Tree ◽  
Riesz Means ◽  
Homogeneous Trees

Abstract Let 𝕋 be a homogeneous tree. We prove that if f ∈ Lp (𝕋), 1 ≤ p ≤ 2, then the Riesz means Sz R (f) converge to f everywhere as R → ∞, whenever Re z > 0.

scholarly journals Large Deviations for Radial Random Walks on Homogeneous Trees

2007 ◽  
Vol 187 ◽  
pp. 75-90
Author(s):  
Kanji Ichihara
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Large Deviations ◽  
Random Walks ◽  
Large Deviation ◽  
Rate Function ◽  
Homogeneous Tree ◽  
Homogeneous Trees ◽  
Radial Random Walks

AbstractDonsker-Varadhan’s type large deviation will be discussed for the pinned motion of a radial random walk on a homogeneous tree. We shall prove that the rate function corresponding to the large deviation is associated with a new Markov chain constructed from the above random walk through a harmonic transform based on a positive principal eigenfunction for the generator of the random walk.

scholarly journals On the Boundary Dieudonné–Pick Lemma

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1108
Author(s):  
Olga Kudryavtseva ◽  
Aleksei Solodov
Keyword(s):  
Fixed Point ◽  
Interior Point ◽  
Sharp Estimate ◽  
General Boundary ◽  
Extremal Functions ◽  
Angular Derivative ◽  
Additional Information ◽  
Carathéodory Theorem ◽  
Class Of Functions ◽  
Boundary Fixed Point

The class of holomorphic self-maps of a disk with a boundary fixed point is studied. For this class of functions, the famous Julia–Carathéodory theorem gives a sharp estimate of the angular derivative at the boundary fixed point in terms of the image of the interior point. In the case when additional information about the value of the derivative at the interior point is known, a sharp estimate of the angular derivative at the boundary fixed point is obtained. As a consequence, the sharpness of the boundary Dieudonné–Pick lemma is established and the class of the extremal functions is identified. An unimprovable strengthening of the Osserman general boundary lemma is also obtained.

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