Analysis on Trees with Nondoubling Flow Measures

Levi, Matteo; Santagati, Federico; Tabacco, Anita; Vallarino, Maria

doi:10.1007/s11118-021-09957-6

Cited by 6 publications

(17 citation statements)

References 24 publications

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“…What is remarkable here, with respect to the results of Section 2, is that flow measures are not bounded above nor below and the tree is not required to have bounded degree. The latter statement implies (see [12,Corollary 2.3]) that the upcoming result applies also to flow measures which are not even locally doubling.…”

Section: Flow Measures On Treesmentioning

confidence: 78%

“…Flows have remarkable properties from the harmonic analysis point of view. Indeed, in [7] the authors develop a Calderón-Zygmund theory on a homogeneous tree endowed with a particular flow and, in [12], this theory is adapted to general trees endowed with any locally doubling flow measure. We define the difference operator acting on functions f : T → C as…”

Section: Flow Measures On Treesmentioning

confidence: 99%

“…Surprisingly, in the last section we are able to prove a global L p -Poincaré inequality for quasiconvex sets and for flow measures on infinite trees, a class of measures introduced in [12] (see (8) for the precise definition). This represents a further evidence that these measures, despite being nondoubling, of exponential growth and not positively bounded from below nor from above, are very well behaved with respect to analysis on trees, see also [13].…”

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confidence: 99%

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Poincaré inequalities on graphs

Levi¹,

Santagati²,

Tabacco³

et al. 2022

Preprint

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We prove local L p -Poincaré inequalities, p ∈ [1, ∞], on quasiconvex sets in infinite graphs endowed with a family of locally doubling measures, and global L p -Poincaré inequalities on connected sets for flow measures on trees. We also discuss the optimality of our results. Introduction and notationPoincaré inequalities have played an important role in analysis on noncompact manifolds, Lie groups, infinite graphs, and more generally on metric spaces. A complete overview of the literature on this subject would go out of the scope of this paper, so we refer the reader to [6,8] and the references therein.While the key role of these inequalities is largely understood, and their validity is considered a natural geometric assumption in the context of analysis in metric spaces, on the other hand, quite surprisingly, in the discrete setting there is a lack of concrete examples of graphs fulfilling these inequalities in the literature. In this note we focus on infinite graphs and we prove that some concrete common families of measures and graphs actually fulfil the desired property of supporting Poincaré-type inequalities.(Local) Poincaré inequalities combined with the (local) doubling condition, are a standard tool to obtain (local) Harnack inequalities both in continuous and discrete settings (see [4,5,14,16]). Applications of the Poincaré inequalities discussed in this note in such direction will be object of further investigation in [11].Let X be an infinite, locally finite, connected, and undirected graph. We identify X with its set of vertices and we write x ∼ y whenever x, y ∈ X are neighbors, namely, when they are connected by an edge. We denote by deg(x) the number of neighbors of x. We say that the graph X has bounded degree b + 1 if deg(x) ≤ b + 1, for some b ≥ 1 and every x ∈ X. A path of length n ∈ N connecting two vertices x and y is a sequence {x 0 , x 1 , . . . , x n } ⊂ X, with no repeated vertices, such that x 0 = x, x n = y, and x i ∼ x i+1 for every i = 0, . . . , n − 1. The distance d(x, y) is defined as the minimum of the lengths of the paths connecting x and y. For any x ∈ X and r ≥ 0, the ball of radius r and center x is B r (x) = {y ∈ X : d(x, y) ≤ r}. For every subset E of X the diameter of E is diam(E) = sup{d(x, y) : x, y ∈ E}.

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Section: Flow Measures On Treesmentioning

confidence: 78%

Section: Flow Measures On Treesmentioning

confidence: 99%

mentioning

confidence: 99%

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Poincaré inequalities on graphs

Levi¹,

Santagati²,

Tabacco³

et al. 2022

Preprint

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“…In the last years the study of harmonic analysis in the homogeneous trees has turned to take great interest. Heat and Poisson semigroups ( [24] and [31]), Poincaré and Hardy inequalities [5], Hardy and BMO spaces ( [3,4,7]), nondoubling flow measures ( [25,26]), uncertainty principles ( [15]), Carleson measures ( [12]), special multipliers ( [8]) and maximal functions ( [14,18,28,29]) are some of the topics that have being recently studied in this setting.…”

Section: Introductionmentioning

confidence: 99%

“…To conclude this introduction we want to mention that harmonic analysis related to flow Laplacian on X has been developed recently in [3,4,5,25,26]. Main definitions and results about trees with nondoubling flow measures can be found in [26]. On X the canonical flow measure λ is defined by…”

Section: Introductionmentioning

confidence: 99%

Pointwise convergence of the heat and subordinates of the heat semigroups associated with the Laplace operator on homogeneous trees and two weighted $L^p$ maximal inequalities

Alvarez-Romero¹,

Barrios²,

Betancor³

2022

Preprint

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In this paper we consider the heat semigroup {Wt}t>0 defined by the combinatorial Laplacian and two subordinated families of {Wt}t>0 on homogeneous trees X. We characterize the weights u on X for which the pointwise convergence to initial data of the above families holds for every f ∈ L p (X, µ, u) with 1 ≤ p < ∞, where µ represents the counting measure in X . We prove that this convergence property in X is equivalent to the fact that the maximal operator on t ∈ (0, R), for some R > 0, defined by the semigroup is bounded from L p (X, µ, u) into L p (X, µ, v) for some weight v on X.

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Riesz Transform for a Flow Laplacian on Homogeneous Trees

Levi

Martini

Santagati

et al. 2023

J Fourier Anal Appl

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We prove the $$L^p$$ L p -boundedness, for $$p \in (1,\infty )$$ p ∈ ( 1 , ∞ ) , of the first order Riesz transform associated to the flow Laplacian on a homogeneous tree with the canonical flow measure. This result was previously proved to hold for $$p \in (1,2]$$ p ∈ ( 1 , 2 ] by Hebisch and Steger, but their approach does not extend to $$p>2$$ p > 2 as we make clear by proving a negative endpoint result for $$p = \infty $$ p = ∞ for such operator. We also consider a class of “horizontal Riesz transforms” corresponding to differentiation along horocycles, which inherit all the boundedness properties of the Riesz transform associated to the flow Laplacian, but for which we are also able to prove a weak type (1, 1) bound for the adjoint operators, in the spirit of the work by Gaudry and Sjögren in the continuous setting. The homogeneous tree with the canonical flow measure is a model case of a measure-metric space which is nondoubling, of exponential growth, does not satisfy the Cheeger isoperimetric inequality, and where the Laplacian does not have spectral gap.

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Analysis on Trees with Nondoubling Flow Measures

Cited by 6 publications

References 24 publications

Poincaré inequalities on graphs

Poincaré inequalities on graphs

Pointwise convergence of the heat and subordinates of the heat semigroups associated with the Laplace operator on homogeneous trees and two weighted $L^p$ maximal inequalities

Riesz Transform for a Flow Laplacian on Homogeneous Trees

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