A Theorem of Roe and Strichartz on homogeneous trees

Pratyoosh Kumar; Sumit Kumar Rano

REVIEW 3 major objections 4 minor 15 references

A Theorem of Roe and Strichartz on homogeneous trees

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A weak-$L^2$ bound on every Laplacian iterate forces a homogeneous-tree function to be an eigenfunction.

desk verdict A plausible and genuinely new tree analogue of Roe/Strichartz, with valuable sharpness examples, but the central nilpotency-removal step is deferred to a previous paper and must be spelled out before the main theorems are proved. read the letter →

arxiv 1908.05998 v1 pith:AFVS5NN4 submitted 2019-08-16 math.FA

Pratyoosh Kumar , Sumit Kumar Rano This is my paper · ORCID

classification math.FA MSC 43A8539A1220E08

keywords homogeneoustreesLaplacianspectrumeigenfunctioncharacterizationweakLpspacessphericalFouriertransformPoissonRoe-Strichartztheorem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

On a homogeneous tree, the natural Laplacian has an elliptic spectrum, and the classical boundedness test for eigenfunctions---uniformly bounded iterates---fails: sums of two spherical eigenfunctions with equal eigenvalue modulus satisfy the uniform bound without being eigenfunctions. The paper proves that the correct substitute is weak-$L^2$ boundedness. If a measurable function $f$ on the tree satisfies $\|L^k f\|_{L^{2,\infty}(X)}\le M|\gamma(z)|^k$ for every integer $k$, for some spectral parameter $z\in\mathbb{R}\setminus(\tau/2)\mathbb{Z}$, then $Lf=\gamma(z)f$, and $f$ is the Poisson transform of an $L^2$ function on the boundary. For $1

What carries the argument

The load-bearing object is the spherical Fourier transform on the tree, which turns radial rapidly decreasing functions into even, $\tau$-periodic holomorphic functions on the spectral strip $S_p=\{z:|\Im z|\le|\delta_p|\}$. The paper proves, in an appendix, that this transform is a topological isomorphism from $S_p(X)^\#$ onto $H(S_p)^\#$, using the Abel-transform isomorphism for radial Schwartz spaces. Through this dictionary, the distributional equation $LT_k=\lambda T_{k+1}$ becomes a pointwise identity $\widehat T_0=e^{ik\theta}(\gamma(\alpha)/\gamma(\cdot))^k\widehat T_k$; letting $k\to\pm\infty$ forces the Fourier transform of $T_0$ to vanish away from the eigenvalue $\pm\alpha$, and a local estimate on $(\gamma(\alpha)-\gamma(s))^{N+1}\varphi$ reduces the eigen-distribution claim to $N=0$. This is what converts boundedness of Laplacian iterates into exact eigenfunctionhood.

What would settle it

On a tree with a fixed $q\ge 2$, compute the two-sided sequence $\|L^k\varphi_s\|_{L^{2,\infty}}$ for two real spectral parameters $s$ with $\gamma(s)\neq\gamma(z)$, and check whether any such sequence is bounded by $M|\gamma(z)|^k$ for all $k\in\mathbb{Z}$; Theorem A predicts that the ratio $|\gamma(s)/\gamma(z)|^k$ eventually destroys the bound, so an explicit counterexample would refute the theorem.

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Extended reading notes

Core claim

The paper's central claim is Theorem A: for a measurable function $f$ on a homogeneous tree $X$ and $z\in\mathbb{R}\setminus(\tau/2)\mathbb{Z}$, if $\|L^k f\|_{L^{2,\infty}(X)}\le M|\gamma(z)|^k$ for all $k\in\mathbb{Z}$, then $Lf\equiv\gamma(z)f$; moreover $f\equiv P_zF$ for some $F\in L^2(\Omega)$, where $P_z$ is the Poisson transform and $\Omega$ is the boundary. Theorem B is the $L^{p'}$ analogue: for $1<p<2$, weak-$L^{p'}$ boundedness of the forward Laplacian iterates at $z=n\tau+i\delta_{p'}$ forces $Lf=\gamma(z)f$, and weak-$L^{p'}$ boundedness of the inverse iterates at $z=(2n+1)\tau/2+i\delta_{p'}$ does the same. The paper also shows these statements are sharp: at the excluded boundary points $\tau/2\mathbb{Z}$ no nonzero $L^{2,\infty}$ eigenfunction exists, the weak-type norms cannot be replaced by stronger Lorentz norms, and both-sided iteration is essential in Theorem A. The same boundedness criterion is worked out on $\mathbb{Z}$, where it yields polynomial-times-exponential eigenfunctions.

Load-bearing premise

The proof depends on the assumption that the dictionary between radial functions on the tree and analytic functions on the spectral strip is exact all the way to the strip's boundary; if that dictionary loses information at the boundary, the estimates that single out one eigenvalue no longer follow.

Editorial extensions

If this is right

Every function satisfying the two-sided weak-$L^2$ growth condition is automatically a Laplacian eigenfunction and, in particular, a Poisson transform of an $L^2$ boundary function.
The uniform $L^\infty$ version of the theorem is false on homogeneous trees; the obstruction is exactly the part of the elliptic spectrum lying on the unit circle, so weak-type bounds are the correct replacement for uniform boundedness.
The excluded boundary parameters $\tau/2\mathbb{Z}$ cannot be added to Theorem A: no nonzero $L^{2,\infty}$ eigenfunction exists there.
For $1<p<2$, one-sided weak-$L^{p'}$ bounds characterize eigenfunctions precisely at the two boundary points of the $L^{p'}$ spectrum; inside the spectrum, explicit non-eigenfunction solutions exist.
On $\mathbb{Z}$, the same boundedness philosophy yields polynomial-times-exponential eigenfunctions, matching the original sine-function characterization in the continuous limit.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same two-sided spectral-support argument should transfer to other graphs with a spherical Fourier transform and a Paley-Wiener theorem, such as distance-regular graphs; the key estimate would again be the decay of $|\gamma(\alpha)/\gamma(\cdot)|^k$ outside the eigenvalue.
Because homogeneous trees are discrete models of hyperbolic space, the paper suggests that the correct Roe-Strichartz statement on hyperbolic spaces is weak-$L^2$ rather than uniform boundedness, and that the known failure there reflects the elliptic spectrum rather than an incidental pathology.
One testable extension is to finite truncations of trees: if the constant $M$ in Theorem A can be chosen independent of truncation size, the characterization would pass to limits of finite graphs and yield a spectral gap criterion.
The $\mathbb{Z}$ result suggests an analogous statement for $\mathbb{Z}^d$ with sublinear growth of the constants, completing the discrete Euclidean side of the analogy.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things you should know. First, this is the first Roe/Strichartz-type characterization on homogeneous trees, and the sharpness examples in Section 4 are genuinely useful. Second, the proof as written has a real gap: Lemma A establishes only (L−γ(α))^{N+1}T0=0, and the passage to (L−γ(α))T0=0 is dismissed with “using the same argument given in [12] one can easily prove”. That step is load-bearing, not cosmetic. The stress-test note is right on target. The hypotheses permit superpositions of spherical functions with equal |γ|, as the paper's own counterexamples show, so the spectral-support argument alone cannot rule out generalized eigenfunctions. You need the uniform bound in k plus the finite-order support to force N=0, and the authors do not supply that argument for the discrete tree setting. Lemma B1 has the same omission. This is the main soft spot, and it is serious enough that the main theorems are not fully proved as written.

What the paper does well: the overall strategy is smart and the right one; the endpoint statements in Theorems A and B are new; the weak-Lp spaces are the natural scale here; and Section 4 gives concrete counterexamples showing the endpoint assumptions cannot simply be weakened. The appendix proves the Paley-Wiener isomorphism for Sp(X)# onto H(Sp)#, which is a real piece of work and is more than many papers in this area include.

The other soft spots are real but secondary. In Lemma B1, some of the displayed estimates look shaky—especially the first case in the V_k^c argument, where |γ(z)| is compared to the imaginary-axis value without properly accounting for the real part. The notation around δ_r and q^{1/r} is also loose. These may be fixable, but they need a careful rewrite. The final Poisson-representation conclusion is imported from the authors' own [11]; citing yourself is not a flaw, but a referee needs to verify that [11] actually contains the required theorem and is in a usable form. If [11] already proves the main Theorems A and B, the novelty here would shrink; from this text alone I cannot tell.

Overall: this is a paper worth engaging with, not a desk reject. The core idea is right, the sharpness section is good, and the gaps look repairable. Send it to peer review with a request for major revision: the authors should either prove the nilpotency-removal step in full or state it as a separate lemma with a complete proof, and they should clean up the estimates in Lemma B1. Once that is done, I would expect the main results to hold up.

Referee Report

3 major / 4 minor

Summary. The paper proves a Roe-Strichartz theorem for homogeneous trees. Theorem A states that if f is measurable on a homogeneous tree and the scaled iterates |γ(z)|^{-k}L^k f are uniformly bounded in weak L^2 for every k∈Z, with z real and not in (τ/2)Z, then Lf=γ(z)f and f is a Poisson transform of an L^2 boundary function. Theorem B gives the analogous one-sided conclusions at the two boundary points of the L^{p'} spectrum for 1<p<2, using weak L^{p'} bounds for positive (or negative) k. Section 4 discusses sharpness, including counterexamples built from superpositions of spherical functions of equal modulus, and a separate statement for the integer line Z. An appendix proves the Paley-Wiener isomorphism between the p-Schwartz space of radial functions and the holomorphic space H(S_p)#.

Significance. If the proof is completed, the result is a genuine and structurally interesting discrete analogue of the Roe-Strichartz theorem, complementing the earlier symmetric-space results of [12] and [13]. The paper has two clear strengths: explicit sharpness counterexamples showing why L∞ boundedness must be replaced by weak L^p bounds, and a self-contained appendix proving the spherical Fourier transform isomorphism for p∈(1,2]. The main defect is that the central proof defers a load-bearing step to an earlier paper on symmetric spaces, without supplying the tree-specific argument; as written, the proof does not rule out generalized eigenfunctions. The overall claim is plausible and the gap appears fixable, but it is substantive rather than cosmetic.

major comments (3)

[Section 3, Lemma A (after Eq. (3.11))] The proof of part (a) for radial distributions establishes only the nilpotent equation (L−γ(α))^{N+1}T0=0, displayed as (3.10)-(3.11). The sentence 'Using the same argument given in [12] one can easily prove that (L−γ(α))T0=0' is exactly the point where the proof must rule out Jordan blocks, and no argument is supplied. This step is load-bearing: the boundedness hypotheses are compatible with superpositions of spherical functions having equal modulus, as the counterexamples in Section 4 and Lemma B1(c) show, so the spectral-support argument by itself cannot distinguish a generalized eigenfunction from an eigenfunction. The displayed estimate (3.12) is used only with N=10m+1; it does not provide a bound for N=0. The paper needs a complete proof, in the homogeneous-tree setting, that (L−γ(α))^{N+1}T0=0 together with the stated distribution bounds implies (L−γ(α))T0=0; citing the symmetric-space result is not sufficient because the local structure of the spherical function and the Paley-Wiener space differ.
[Section 3, Lemma B1] The same deferred step occurs in the proof of Lemma B1: after showing that sup_{z∈S_p^+} F_k(z) tends to 0, the proof states 'Using the same argument given in [12] one can easily prove that N=0.' This is the only passage from (γ(z0)−γ(z))^{N+1}\hat T0=0 to the eigenfunction identity LT0=|λ|T0, and it is essential for Theorem B. In addition, the derivative estimates labelled (i) and (ii) after (3.14) are asserted without proof; the limiting argument needs those bounds to be uniform in k and z, with constants independent of k. Please supply the missing nilpotency-removal argument and prove or properly reference the derivative estimates.
[Section 3, Lemma B1, definition of V_k] The set V_k is defined as {z∈S_p^+ : |Re z|<(k^{1/4}log q)^{-1}, δ_p−log(1+1/k^{1/6})(log q)^{-1}<Im z≤δ_p}. Since δ_p=1/p−1/2 is negative for 1<p<2, this set is empty as written, and the following case distinction 'if 0≤Im z≤δ_p−...' is impossible. The intended quantity is evidently δ_{p'}=−δ_p, the upper boundary of S_p^+. This is a substantive notation error in the proof of the main p-dependent lemma; please correct it and verify that the estimates (3.15)-(3.16) hold on the intended set.

minor comments (4)

[Abstract and Section 1] In the statement of Roe's theorem, 'for all n and x' should be 'for all k and x'; the sequence is indexed by k.
[Section 4, item (1)] The sharpness claim for z∈(τ/2)Z is not fully established: the argument shows that an eigenfunction at such z cannot lie in L^{2,∞}, but it does not by itself produce a function satisfying the boundedness hypothesis that fails to be an eigenfunction. Please state the intended logical role of this observation.
[Section 4.1, Theorem 4.1] Theorem 4.1 for the integer line is stated without proof; if it is included as a result, a proof or a precise reference should be provided rather than the remark that it can be proved by the similar argument of Roe.
[Section 3, Lemma B1] Because λ may have a nonzero argument, the proof silently removes a phase by replacing T_k with e^{-ikθ}T_k, where λ=|λ|e^{iθ}; this reduction should be made explicit, since the displayed Fourier relations otherwise appear to assume λ=γ(iδ_{p'}).

Circularity Check

3 steps flagged · score 4.0 of 10

Central eigenfunction conclusion rests on two deferred 'same argument as [12]' steps and a self-cited Poisson representation; the spectral-support core is otherwise self-contained.

self citation load bearing [Section 3, Lemma A, after Eq. (3.12)]
"Using the same argument given in [12] one can easily prove that (L − γ(α))T0 = 0."

The preceding spectral-support argument establishes only (L−γ(α))^{N+1}T0=0 (Eqs. 3.10–3.11). The genuine eigenfunction identity (L−γ(α))T0=0 is the core conclusion of Lemma A and of Theorem A, but instead of being proved it is deferred to [12], whose first author overlaps with the present first author. No tree-specific argument is supplied for removing the nilpotent part, and the counterexamples in Section 4 show that the spectral-support conclusion alone does not force T0 to be an eigenfunction. Thus the decisive final step of the central claim is load-bearing on an overlapping-author citation.
self citation load bearing [Section 3, Lemma B1, after Eqs. (3.13)–(3.14)]
"Using the same argument given in [12] one can easily prove that N = 0."

Lemma B1 proves (γ(z0)−γ(z))^{N+1} T0-hat = 0 for some N, but the conclusion of the lemma is the eigenfunction identity, which is the case N=0. That reduction is again not carried out; it is asserted to follow from the same argument as in [12], a paper by Kumar, Ray, and Sarkar whose first author is the present first author and which treats Laplace–Beltrami operators rather than homogeneous trees. This is an essential step for Theorem B, not a cosmetic remark, so the proof's central deduction depends on a self-citation.

1 more flagged steps

self citation load bearing [Section 1, after Theorem B]
"It was proved in [11] that any weak Lp eigenfunction of the Laplacian of X can be represented by the Poisson transform of a Lp function on the boundary. Therefore the conclusion of these theorems is more precise than that of Theorem 1.1."

The 'In particular, there exists F ∈ L^p'(Ω) such that f ≡ P_z F' statements in Theorems A and B are not proved in this paper; they are imported from [11], an arXiv preprint by the same two authors. This makes the stronger representation conclusion of the main theorems dependent on an overlapping-author self-citation. It is not used to prove Lf=γ(z)f, so it is a secondary load-bearing step rather than the source of the main eigenfunction identity.

full rationale

The paper is not circular in the definitional or fitted-parameter sense: no constant is fitted to the data it predicts, and Theorem A/B are not obtained by renaming an assumption. The spectral-support argument in Lemmas A and B1 is a substantial, mostly self-contained derivation using the spherical Fourier transform, and the appendix supplies the needed isomorphism theorem (Theorem 5.1) from the independent Cowling–Meda–Setti Abel-transform result. However, the final and essential step that converts the generalized eigenfunction identity (L−γ(α))^{N+1}T0=0 into the genuine eigenfunction identity (L−γ(α))T0=0 is not proved; in both Lemma A and Lemma B1 it is deferred with 'Using the same argument given in [12]'. Since [12] shares the present first author and concerns Laplace–Beltrami operators on symmetric spaces rather than homogeneous trees, the central claim's decisive step is load-bearing on an overlapping-author citation. The 'in particular' Poisson-representation conclusions in Theorems A and B are likewise imported from the same-two-author preprint [11]. These are self-citation burdens, not definitional equivalences; the main eigenfunction identity still has independent analytic content, so the appropriate score is moderate rather than maximal.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted and no new entities are invented. The central claim rests on standard harmonic analysis on homogeneous trees, the spherical Fourier isomorphism theorem, the known formula for the L^p spectrum, and the Poisson representation theorem imported from the authors' earlier work.

assumptions (4)

standard math The spherical Fourier transform is a topological isomorphism from S_p(X)# onto H(S_p)# for 1<p≤2 (Theorem 5.1).
Used throughout Section 3 to pass between distributions and holomorphic functions. Proved in the appendix, but the proof relies on Cowling et al.'s Abel transform isomorphism.
domain assumption The L^p spectrum of L is the image of S_p under gamma, given by the ellipse (2.5).
Quoted from [3, page 4275]; used in Lemma A to know that |z0| in [1-b,1+b] is an eigenvalue modulus.
domain assumption Any weak L^p eigenfunction of L on X can be represented as the Poisson transform of an L^p function on the boundary.
Cited to [11] by the same authors; used in the 'in particular' conclusions of Theorems A and B. The paper does not prove it.
standard math The Poisson transform P_z maps boundary functions to eigenfunctions with L P_z eta = gamma(z) P_z eta, and the spherical functions phi_z are uniformly bounded for z in S_1.
Used in the counterexamples and sharpness discussion; standard in the tree harmonic analysis literature.

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Cite this review

Pith. "Pith review of A Theorem of Roe and Strichartz on homogeneous trees." pith.science (2026). https://pith.science/paper/AFVS5NN4

@misc{pith2026190805998,
  author       = {Pith},
  title        = {Pith review of: A Theorem of Roe and Strichartz on homogeneous trees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AFVS5NN4}},
  note         = {Machine review of arXiv:1908.05998}
}

abstract

In 1980, J. Roe proved that if $\{f_{k}\}_{k\in\mathbb{Z}}$ is doubly infinite sequence of functions in $\mathbb{R}$ which is uniformly bounded and satisfies $(df_{k}/dx)=f_{k+1}$ for all $k\in\mathbb{Z}$ then $f_{0}(x)=a\sin(x+\theta)$ for some $a,\theta\in\mathbb{R}$. Later in 1993 Strichartz suitably extended the above result to $\mathbb{R}^n$. In this article we prove a version of their result for homogeneous trees.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[12]

Kumar, S

P. Kumar, S. K. Ray and R. P. Sarkar, Characterization of almost Lp-eigenfunctions of the Laplace-Beltrami operator, Trans. Amer. Math. Soc. 366 (2014), no. 6, 3191-3225

work page 2014
[11]

Analysis of $L^p$-type estimates of Poisson transform on Homogeneous Trees

P. Kumar and S. K. Rano, Analysis of Lp-type estimates of Poisson transform on homogeneous trees , arXiv:1808.09659

work page Pith review arXiv
[13]

S. K. Ray and R. P. Sarkar, A theorem of Roe and Strichartz for Riemannian symmetric spa ces of non- compact type, Int. Math. Res. Not. IMRN (2014), no. 5, 1273-1288

work page 2014
[1]

Betori, J

W. Betori, J. Faraut and M. Pagliacci, An inversion formula for the Radon transform on trees , Math. Z. 201 (1989), no. 3, 327-337

work page 1989
[2]

Cowling, S

M. Cowling, S. Meda and A. Setti, An overview of harmonic analysis on the group of isometries o f a homogeneous tree, Exposition. Math. 16 (1998), no. 5, 385-423

work page 1998
[3]

Cowling, S

M. Cowling, S. Meda and A. Setti, Estimates for functions of the Laplace operator on homogene ous trees, Trans. Amer. Math. Soc. 352 (2000), no. 9, 4271-4293

work page 2000
[4]

Bianca Di Blasio, Paley-Wiener type theorems on harmonic extensions of H-typ e groups , Monatsh. Math. 123 (1997), no. 1, 21-42

work page 1997
[5]

Fig` a-Talamanca and C

A. Fig` a-Talamanca and C. Nebbia, Harmonic analysis and representation theory for groups act ing on homogeneous trees, London Math. Soc. Lecture Notes Series, 162 (Cambridge University Press, Cambridge, 1991)

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[6]

Fig` a-Talamanca and M

A. Fig` a-Talamanca and M. Picardello, Spherical functions and harmonic analysis on free groups , J. Funct. Anal. 47 (1982), no. 3, 281-304

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Fig` a-Talamanca and M

A. Fig` a-Talamanca and M. Picardello, Harmonic analysis on free groups , Lecture Notes in Pure and Applied Mathematics, 87 (Dekker, New York, 1983)

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Grafakos, Classical Fourier analysis, Third edition, Graduate Texts in Mathematics, 249 (Springer, New York, 2014)

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Howard, A note on Roe’s characterization of the sine function , Proc

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R. Howard, M. Reese Characterization of eigenfunctions by boundedness condit ions, Canad. Math. Bull. 35 (1992), no. 2, 204-213

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R. S. Strichartz, Characterization of eigenfunctions of the Laplacian by bou ndedness conditions , Trans. Amer. Math. Soc. 338 (1993), no. 2, 971-979. Department of Mathematics, Indian Institute of Technology Guw ahati, 781039, India. E-mail: pratyoosh@iitg.ac.in and s.rano@iitg.ac.in

1993

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [12]

Kumar, S

P. Kumar, S. K. Ray and R. P. Sarkar, Characterization of almost Lp-eigenfunctions of the Laplace-Beltrami operator, Trans. Amer. Math. Soc. 366 (2014), no. 6, 3191-3225

work page 2014

[2] [11]

Analysis of $L^p$-type estimates of Poisson transform on Homogeneous Trees

P. Kumar and S. K. Rano, Analysis of Lp-type estimates of Poisson transform on homogeneous trees , arXiv:1808.09659

work page Pith review arXiv

[3] [13]

S. K. Ray and R. P. Sarkar, A theorem of Roe and Strichartz for Riemannian symmetric spa ces of non- compact type, Int. Math. Res. Not. IMRN (2014), no. 5, 1273-1288

work page 2014

[4] [1]

Betori, J

W. Betori, J. Faraut and M. Pagliacci, An inversion formula for the Radon transform on trees , Math. Z. 201 (1989), no. 3, 327-337

work page 1989

[5] [2]

Cowling, S

M. Cowling, S. Meda and A. Setti, An overview of harmonic analysis on the group of isometries o f a homogeneous tree, Exposition. Math. 16 (1998), no. 5, 385-423

work page 1998

[6] [3]

Cowling, S

M. Cowling, S. Meda and A. Setti, Estimates for functions of the Laplace operator on homogene ous trees, Trans. Amer. Math. Soc. 352 (2000), no. 9, 4271-4293

work page 2000

[7] [4]

Bianca Di Blasio, Paley-Wiener type theorems on harmonic extensions of H-typ e groups , Monatsh. Math. 123 (1997), no. 1, 21-42

work page 1997

[8] [5]

Fig` a-Talamanca and C

A. Fig` a-Talamanca and C. Nebbia, Harmonic analysis and representation theory for groups act ing on homogeneous trees, London Math. Soc. Lecture Notes Series, 162 (Cambridge University Press, Cambridge, 1991)

work page 1991