The Lusin area function and local admissible convergence of harmonic
functions  on homogeneous trees

Atanasi, Laura; Picardello, Massimo A.

doi:10.1090/s0002-9947-07-04433-9

Cited by 12 publications

(26 citation statements)

References 17 publications

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“…Our approach adapts to this discrete environment ideas developed in [10,12] for the product of discs and of rankone symmetric spaces (see also [8]). The argument overcomes several complications arising from the use of discrete difference equations instead of the classical identities for the Green function and the Laplacian, and extends the results of [1] valid for one tree. The converse implication, finiteness of this area function on a product ⇒ existence of admissible limits almost everywhere, cannot hold in full generality in a cartesian product (see Proposition 2.1; for hints on variants of the area theorem suitable to prove the converse implication see Remark 2.2).…”

Section: Introductionmentioning

confidence: 55%

“…This equivalence has been established for rank-one symmetric spacess in [9] and for infinite homogeneous trees in [1,6,11] and, with a more probabilistic argument, in [14]. The aim of this paper is to prove an area theorem for product of trees, of the type: existence of admissible limits ⇒ finiteness of the area function almost everywhere.…”

Section: Introductionmentioning

confidence: 94%

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An Area Theorem for Joint Harmonic Functions on the Product of Homogeneous Trees

Atanasi

Picardello

2021

Potential Anal

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For harmonic functions v on the disc, it has been known for a long time that non-tangential boundedness a.e.is equivalent to finiteness a.e. of the integral of the area function of v (Lusin area theorem). This result also hold for functions that are non-tangentially bounded only in a measurable subset of the boundary, and has been extended to rank-one hyperbolic spaces, and also to infinite trees (homogeneous or not). No equivalent of the Lusin area theorem is known on higher rank symmetric spaces, with the exception of the degenerate higher rank case given by the cartesian product of rank-one hyperbolic spaces. Indeed, for products of two discs, an area theorem for jointly harmonic functions was proved by M.P. and P. Malliavin, who introduced a new area function; non-tangential boundedness a.e. is a sufficient condition, but not necessary, for the finiteness of this area integral. Their result was later extended to general products of rank-one hyperbolic spaces by Korányi and Putz. Here we prove an area theorem for jointly harmonic functions on the product of a finite number of infinite homogeneous trees; for the sake of simplicity, we give the proofs for the product of two trees. This could be the first step to an area theorem for Bruhat–Tits affine buildings, thereby shedding light on the higher rank continuous set-up.

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Section: Introductionmentioning

confidence: 55%

Section: Introductionmentioning

confidence: 94%

An Area Theorem for Joint Harmonic Functions on the Product of Homogeneous Trees

Atanasi

Picardello

2021

Potential Anal

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“…A generalization of the theorem of Spencer [42] to several variables was obtained in Stein [43]. Since then, criteria on existence of non-tangential boundary limits of harmonic functions in many different contexts, in terms of non-tangential boundedness or one-side non-tangential boundedness or finiteness of area integrals have been intensively studied; see, for example, [1][2][3][4]7,[14][15][16][17][18][19][20][21][22]24,[32][33][34][35][36][37]39] and [46].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…1) whenever the right hand side above is finite.Proof. Let χ Γ h α (0) be the characteristic function of Γ h α (0), i.e., χ Γ h α (0) (x, y) = 1 if |x| < αy and 0 < y < h,…”

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confidence: 99%

Boundary Values of Generalized Harmonic Functions Associated with the Rank-One Dunkl Operator

sci¹

2020

ATA

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We consider the local boundary values of generalized harmonic functions associated with the rank-one Dunkl operator D in the upper half-plane R 2 + = R × (0, ∞), where (D f)(x) = f (x) + (λ/x)[ f (x) − f (−x)] for given λ ≥ 0. A C 2 function u in R 2 + is said to be λ-harmonic if (D 2 x + ∂ 2 y)u = 0. For a λ-harmonic function u in R 2 + and for a subset E of ∂R 2 + = R symmetric about y-axis, we prove that the following three assertions are equivalent: (i) u has a finite non-tangential limit at (x, 0) for a.e. x ∈ E; (ii) u is non-tangentially bounded for a.e. x ∈ E; (iii) (Su)(x) < ∞ for a.e. x ∈ E, where S is a Lusin-type area integral associated with the Dunkl operator D.

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“…For further tips of the iceberg, see e.g. Casadio Tarabusi, Cohen, Korányi and Picardello [7], Rigoli, Salvatori and Vignati [18], Cohen, Colonna and Singman [9], Atanasi and Picardello [3] or Casadio Tarabusi and Figà-Talamanca [8], and the references given there.…”

Section: A Table Of Correspondencesmentioning

confidence: 99%

Moments of Riesz measures on Poincaré disk and homogeneous tree–A comparative study

Boiko

Woess

2015

Expositiones Mathematicae

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One of the purposes of this paper is to clarify the strong analogy between potential theory on the open unit disk and the homogeneous tree, to which we dedicate an introductory section. We then exemplify this analogy by a study of Riesz measures. Starting from interesting work by Favorov and Golinskii [10], we consider subharmonic functions on the open unit disk, resp. on the homogenous tree. Supposing that we can control the way how those functions may tend to infinity at the boundary, we derive moment type conditions for the Riesz measures. One one hand, we generalise the previous results of [10] for the disk, and on the other hand, we show how to obtain analogous results in the discrete setting of the tree.

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The Lusin area function and local admissible convergence of harmonic functions on homogeneous trees

Abstract: Abstract. We prove admissible convergence to the boundary of functions that are harmonic on a subset of a homogeneous tree by means of a discrete Green formula and an analogue of the Lusin area function.

Cited by 12 publications

References 17 publications

An Area Theorem for Joint Harmonic Functions on the Product of Homogeneous Trees

An Area Theorem for Joint Harmonic Functions on the Product of Homogeneous Trees

Boundary Values of Generalized Harmonic Functions Associated with the Rank-One Dunkl Operator

Moments of Riesz measures on Poincaré disk and homogeneous tree–A comparative study

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