A theorem of Roe and Strichartz on homogeneous trees

Sumit Kumar Rano

doi:10.1515/forum-2020-0358

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A theorem of Roe and Strichartz on homogeneous trees

Sumit Kumar Rano

Published/Copyright: December 1, 2021

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From the journal Forum Mathematicum Volume 34 Issue 1

Abstract

Let 𝔛 be a homogeneous tree and let ℒ be the Laplace operator on 𝔛. In this paper, we address problems of the following form: Suppose that {fk}k∈ℤ is a doubly infinite sequence of functions in 𝔛 such that for all k∈ℤ one has ℒ⁢fk=A⁢fk+1 and ∥fk∥≤M for some constants A∈ℂ, M>0 and a suitable norm ∥⋅∥. From this hypothesis, we try to infer that f0, and hence every fk, is an eigenfunction of ℒ. Moreover, we express f0 as the Poisson transform of functions defined on the boundary of 𝔛.

Keywords: Homogeneous tree; spectrum of Laplacian; eigenfunction of Laplacian; Fourier analysis

MSC 2010: 43A85; 39A12; 20E08

Communicated by Jan Frahm

A Appendix

To make the exposition self-contained, we now prove that the spherical Fourier transform is a topological isomorphism between the p-Schwartz space 𝒮p⁢(𝔛)# and the space ℋ⁢(Sp)# for every p∈(1,2]. It was observed in [2] that the spherical Fourier transform of a finitely supported radial function f on 𝔛 can also be written as

(A.1)f^⁢(z)=∑n∈ℤ𝒜⁢f⁢(n)⁢qi⁢n⁢z,

where 𝒜⁢f denotes the Abel transform of f. In the same paper, Cowling, Meda and Setti proved that for every p∈(1,2] the map f↦𝒜⁢f is a topological isomorphism from 𝒮p⁢(𝔛)# onto q-δp⁢|⋅|⁢Sev⁢(ℤ) (see [2, Theorem 2.5]), where Sev⁢(ℤ) is a Fréchet space of all such even functions F on ℤ for which

λm⁢(F)=supn∈ℤ⁡(1+|n|)m⁢|F⁢(n)|<∞ for all ⁢m∈ℤ+.

More precisely, they proved that f↦𝒜⁢f is a bijection from 𝒮p⁢(𝔛)# onto q-δp⁢|⋅|⁢Sev⁢(ℤ), and for every natural number m≥2 there exists a positive constant C, depending on p and m, such that for all f∈𝒮p⁢(𝔛)#,

(A.2)C-1⁢λm-2⁢(qδp⁢|⋅|⁢𝒜⁢f)≤νp,m⁢(f)≤C⁢λm⁢(qδp⁢|⋅|⁢𝒜⁢f).

We use the above result to prove the following isomorphism theorem.

Theorem A.1.

The map f→f^ is a topological isomorphism from Sp⁢(X)# onto H⁢(Sp)# for every p∈(1,2]. In particular, for every m∈Z+ and for every p∈(1,2], there exist positive constants C1 and C2 such that

(A.3)C1⁢μp,m⁢(f^)≤νp,m+4⁢(f)≤C2⁢max⁡{μp,m+4⁢(f^),μp,0⁢(f^)} for all ⁢f∈𝒮p⁢(𝔛)#.

Proof.

Since the 𝒮2⁢(𝔛)#-isomorphism theorem is already proved in [1, 4], we consider 1<p<2. Suppose that f∈𝒮p⁢(𝔛)#. Then 𝒜⁢f belongs to q-δp⁢|⋅|⁢Sev⁢(ℤ), and hence

λm⁢(qδp⁢|⋅|⁢𝒜⁢f)=supn∈ℤ⁡(1+|n|)m⁢qδp⁢|n|⁢|𝒜⁢f⁢(n)|<∞ for all ⁢m∈ℤ+.

By using the above fact, it follows that for every z∈Sp,

∑n∈ℤ|𝒜⁢f⁢(n)|⁢|qi⁢n⁢z|=∑n∈ℤ|𝒜⁢f⁢(n)|⁢q-n⁢(ℑ⁡z)

=∑n=0∞|𝒜⁢f⁢(n)|⁢q-n⁢(ℑ⁡z)+∑n=1∞|𝒜⁢f⁢(n)|⁢qn⁢(ℑ⁡z)

≤C⁢λ4⁢(qδp⁢|⋅|⁢𝒜⁢f)⁢[∑n=0∞1(1+n)4⁢q-n⁢(ℑ⁡z+δp)+∑n=1∞1(1+n)4⁢qn⁢(ℑ⁡z-δp)]

≤C⁢λ4⁢(qδp⁢|⋅|⁢𝒜⁢f)⁢∑n=1∞1(1+n)4<∞.

Therefore, the infinite series (A.1) converges uniformly on Sp, and consequently f^ is well-defined. The analyticity of f^ on Sp∘ follows directly from the analyticity of the function qi⁢n⁢z, together with the fact that the infinite series (A.1) converges uniformly on every compact subset of Sp∘. In fact, for every m∈ℤ+,

(dmd⁢zm)⁢f^⁢(z)=∑n∈ℤ(i⁢n⁢log⁡q)m⁢𝒜⁢f⁢(n)⁢qi⁢n⁢z for all ⁢z∈Sp∘.

Note that the right-hand side of the above expression extends to a uniformly convergent series on the whole of Sp. By using this fact together with the continuity of the function qi⁢n⁢z, it follows that f^ and all its derivatives extend continuously to Sp. Moreover, using (A.2), we find that for every semi-norm μp,m of ℋ⁢(Sp)# there exists a semi-norm νp,m+4 of 𝒮p⁢(𝔛)# such that

μp,m⁢(f^)≤C⁢νp,m+4⁢(f) for all ⁢f∈𝒮p⁢(𝔛)#.

Conversely, assume g∈ℋ⁢(Sp)#. Then, for all r satisfying p<r≤2, the function g(⋅+iδr) is infinitely differentiable on the real line, with periodicity τ. Hence g has a Fourier series representation of the form g⁢(α)=∑n∈ℤF⁢(n)⁢qi⁢n⁢α, α∈[-τ/2,τ/2), where

F⁢(n)=1τ⁢∫-τ/2τ/2g⁢(α)⁢q-i⁢n⁢α⁢𝑑α

yields the n-th Fourier coefficient of the function g. Our aim is to prove that F∈q-δp⁢|⋅|⁢Sev⁢(ℤ). By applying Cauchy’s integral theorem to the function g, it is easy to verify that for every r∈(p,2] and n∈ℤ,

(A.4)F⁢(n)=1τ⁢∫-τ/2τ/2g⁢(α+i⁢δr)⁢q-i⁢n⁢(α+i⁢δr)⁢𝑑α=1τ⁢∫-τ/2τ/2g⁢(α-i⁢δr)⁢q-i⁢n⁢(α-i⁢δr)⁢𝑑α.

In fact, the first equality in (A.4) can be proved by applying Cauchy’s integral theorem to g on the closed rectangle

Γ⁢(z)={z∈ℂ:ℑ⁡z=0,-τ2≤Re⁡z≤τ2}∪{z∈ℂ:Re⁡z=τ2,0≤ℑ⁡z≤δr}

∪{z∈ℂ:ℑ⁡z=δr,τ2≤Re⁡z≤-τ2}∪{z∈ℂ:Re⁡z=-τ2,δr≤ℑ⁡z≤0}.

Using the identities (A.4) and noting that g is even, one can easily prove that F⁢(-n)=F⁢(n) for all n∈ℕ, that is, F is an even function on ℤ. Integrating by parts the expressions in (A.4) m times and further using the dominated convergence theorem and letting r→p, we have

(1+|n|)m⁢q|n|⁢δp⁢|F⁢(n)|≤C⁢μp,m⁢(g) for all ⁢m∈ℤ+⁢ and all ⁢n∈ℤ∖{0}.

The above expression together with the trivial estimate |F⁢(0)|≤μp,0⁢(g) gives us

(A.5)λm⁢(qδp⁢|⋅|⁢F)≤C⁢max⁡{μp,m⁢(g),μp,0⁢(g)} for all ⁢m∈ℤ+.

Hence there exists a unique f∈𝒮p⁢(𝔛)# such that 𝒜⁢f=F and f^=g. Combining (A.2) and (A.5), we conclude that

νp,m⁢(f)≤C⁢max⁡{μp,m⁢(f^),μp,0⁢(f^)} for all ⁢m≥2.

This completes the proof. ∎

Acknowledgements

The author would like to thank Pratyoosh Kumar and Swagato K. Ray for suggesting this problem and for many useful discussions during the course of this work. The author also wishes to thank Rudra P. Sarkar and an unknown referee for several important suggestions which improved an earlier draft.

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Received: 2020-12-24

Revised: 2021-10-09

Published Online: 2021-12-01

Published in Print: 2022-01-01

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Keywords for this article

Homogeneous tree; spectrum of Laplacian; eigenfunction of Laplacian; Fourier analysis