The dissertation work entitled “Some Aspects of the Poisson Transform on Homogeneous Trees” by Sumit Kumar Rano, a student of the Department of Mathematics, Indian Institute of Technology Guwahati, for the award of the degree of Doctor of Philosophy is certified to have been done under my supervision and this work has not been submitted elsewhere for the degree . First, we characterize all eigenfunctions of the Laplacian on homogeneous trees, which are the Poisson transformation of functions Lp defined on the boundary.
Eigenfunctions of the Laplacian
The most important ingredient in the proof is the norm estimates of matrix coefficients of the representation πz. In this thesis we answer this question by proving a version of Theorem 1.1.3 in the context of homogeneous trees.
Fourier Restriction Theorem
It turns out that the above estimate can also be seen as the limit (2, p0) of the Poisson transformation via the relation. More precisely, the Thomas-Stein theorem can also be proved using duality in the successive limit of the Poisson transform and vice versa.
A Theorem of Roe and Strichartz
A careful analysis of the above counterexample shows that the failure of the Strichartz result on X is mainly due to the spectrum of L. It was also observed in [27] that the failure of the Strichartz result is deeply rooted in the p dependence on the Lp spectrum. of the Laplace-Beltrami operator ∆ on the hyperbolic spaces.
Dynamics of Semigroups Generated by Laplacian
A similar result regarding the chaotic behavior of the heat semigroup and the Dunkl heat semigroup is also known for NA harmonic groups and Euclidean spaces, respectively. In addition, we separately study the dynamic behavior of the semigroups generated by affine functions, namely T(t) = exp{t(aL+b)}, a∈C\ {0}, b∈R and derive a sharp p- depending on range for which the affine semigroup is hypercyclic.
Outline of the Thesis
We will use standard practice to use the letter C for constant, the value of which can change from one. From now on we will write Sp◦ and ∂Sp to denote the usual interior and limit of Sp, respectively.
Homogeneous Trees
Basic Structure
In the above expression, dk represents the normalized Haar measure on K. Writing the above expression as an integral on X, it follows. Statements (2) and (3) follow easily from the definition above, and so we omit the proof. 4): For a finitely supported function f on X, Z.
Boundary and Poisson Kernel
The measure ν on the limit Ω is a G-quasi-invariant probability measure and the Poisson kernel p(g ·o, ω) is defined to be the Radon-Nikodym derivative dν(g−1ω)/dν(ω) which is given explicitly by the formula. From the clear formula above, it follows that the Poisson kernel is a non-negative function on X×Ω, which for any fixed x, takes only many finite values as a function of ω.
Laplacian
Many authors have also adopted the formula L to denote the Laplacian onX (see e.g. [12, page 34]). The Laplacian L is a G-invariant and bounded operator on Lp(X), can be given by convolution to the right of a radial function on X, namely.
Poisson Transform and Eigenfunction
- Conditional Expectations and Martingales
- Poisson Transform
- Eigenfunctions of the Laplacian
- The Spherical Functions
- Spectrum of the Laplacian
Using the above duality, we now extend the definition of the Poisson transform to a martingaleF= (Fn) si. For z ∈C, the elementary spherical function φz is the radial Laplacian eigenfunction normalized by φz(o) = 1. As a consequence of the above lemma, we have the following magnitude estimates for the spherical function φz.
We leave the proofs as they follow easily from the definition of Lorentz norms and the point estimates of φz given in Lemma 2.3.11. Since L moves with both these operators (see Proposition 2.2.7), it follows that f is a radial eigenfunction of the Laplacian with eigenvalue γ(z).
Fourier Transform on Homogeneous Trees
Spherical Fourier Transform
2.4.15) It is known that Sp(X) forms a Fr´echet space with respect to these countable seminorms νm(·) and they are also known as the p-Schwartz spaces for rapidly decreasing functions on X (see e.g. e.g. [5] ). We now aim to characterize the range of the spherical Fourier transform with Sp(X)# as domain. It was observed in [5] that the spherical Fourier transform of a finitely supported radial function f can also be written as.
In the same article, Cowling, Meda and Setti proved that for every p ∈ [1,2] the mapping f → Af is a topological isomorphism from Sp(X)# to q−δp|·|Sev(Z), where Sev( Z) is the space of all even functions on Z such that. 2.4.17) Using the above result as a tool, we are now ready to prove the isomorphism theorem for the spherical Fourier transform on X. Then it is clear that the infinite series (2.4.16) converges uniformly to Sp and consequently ˆf is well defined.
Helgason-Fourier Transform
In this chapter we aim to prove certain results characterizing all eigenfunctions of the Laplacian on homogeneous trees, which are the Poisson transformation of Lp0 functions defined on the boundary. This is mostly done by carefully analyzing the magnitude estimates of the Poisson transform of Lp0 functions when 1< p ≤2. To take a step forward, we now impose some size conditions on these eigenfunctions of the Laplacian to ensure that they are indeed the Poisson transform of Lp0 functions on Ω.
Here it is worth mentioning that these size estimates arise naturally from the behavior of the Poisson transform (of Lp0 functions defined on the boundary) which in turn is a generalization of the behavior of the spherical . In particular, this leads to the fact that PzF is a weak Lp0 eigenfunction of L, which in turn indicates the use of Lp0,∞ as the size condition to characterize the eigenfunctions of Las the Poisson transform of Lp0 functions.
Some Important Results
To prove the main result of this chapter, we also need the following result on the asymptotic behavior of the Poisson transformation. We now state some important properties of the Harish-Chandra c-function that are useful in what follows.
Weak L p Eigenfunctions
To prove the converse of this theorem, we first show that for all F ∈Lp0(Ω) there exists a constant C (independent of F and α) such that. Since the above constant C is independent of n, it follows from Theorem 2.3.7 that the martingaleF is given by the function aLp0(Ω), say F.
The Case p = 2
This difference occurs due to the fact that the constants that appeared in norm estimation (3.3.6) are independent of the parameter z ∈ R+iδp0. On the other hand, the constants that appeared in the norm estimate (3.3.12) depend on the Harish-Chandra c function. These results can also be considered as an application of the magnitude estimates of the Poisson transform proven in Chapter 3.
The key concept involved here is the formulation of the duality relation between the Poisson transform and the Helgason-Fourier transform. Finally, in Section 4.4 we discuss the existence and analyticity of the Helgason-Fourier transform on X .
The Intertwining Operators
This idea was in turn adapted from [25,26] where the authors proved similar results on symmetric spaces and harmonic groups N A. In section 4.2 we recall some important facts about the interpolation operators, which play a role vital in validating our main results. Moreover, they also proved that for some selective values of z, Iz turns out to be an integral operator, which in our context can be expressed in the following way.
The main purpose of this section is to prove certain important facts about the operator Iiδp0 that will be used further.
Fourier Restriction Theorems
Restriction Theorem for the Helgason-Fourier Transform
Now we prove our main results for the Helgason-Fourier transformation on X, which can also be considered as an extension of Theorem 4.3.1. Our proofs are mainly based on the following duality relation: Forz ∈C, a finitely supported function on X and F ∈ K(Ω), . Given the duality relation (4.3.4), it follows that to prove our statement, it suffices to prove that for any p < r < p0,. 4.3.6) We divide this proof into the following steps.
Existence of the Helgason-Fourier Transform
In this chapter we will witness yet another way to characterize the eigenfunctions of the Laplacian L on a homogeneous tree X. As described in Chapter 1, Srichartz's result can also be seen as a representative theorem that characterizes all eigenfunctions of the Laplacian ∆Rn with eigenvalue − 1. Note that the most elementary eigenfunction satisfying the hypothesis of Theorem 5.1.1 is formei.
An obvious analogue of these functions on homogeneous trees are the complex powers of the Poisson kernel, i.e. p1/2+iλ(x, ω), which also act as elementary eigenfunctions of the Laplacian L (see Section 2.3.3, Chapter 2 for details). ). Unlike the L spectrum, which is an elliptic region depending on p, the LZ spectrum is always a linear segment, i.e. [0,2].
Tempered distributions on Homogeneous Trees
Roe’s Theorem for Tempered Distributions
Having gathered all the necessary information, we are now ready to prove our results for Lp-tempered distributions. Again using the fact that L commutes with translations and radialization (see Proposition 2.2.7) and LTk = zTk+1, we have L(R(τxTk)) =zR(τxTk+1) for all k ∈Z. Since the result is already proven for radial L2-relaxed distributions, we have LR(τxT0) =|z0|R(τxT0) for all x∈X.
The main difference from the previous theorem and the classical Euclidean case is that the Lp-tempered distributions act on holomorphic functions. For 1 < p < 2, let {Tk}k∈Z+ be an infinite sequence of Lp-tempered distributions on X satisfying,.
Functions which are Tempered Distributions
Roe’s Results for Almost L p -Functions
Sharpness of the Results
In this section, we will try to establish the sharpness of our main results by answering the following natural questions. In fact, from the relevant discussions we will also get a transparent idea about the formulation of Theorem 5.3.1 and 5.3.2. Since L commutes with both of these operators (see Proposition 2.2.7), it follows that f is a radial eigenfunction of the Laplacian with eigenvalue γ(z).
In light of the radialization technique used in the proof of the previous question, it suffices to prove the following. Using this estimate, we now evaluate the distribution functiondφz(·) of φz as. which is not convergent by the Monotone Convergence Theorem.
In the second part, we separately study the dynamic behavior of the semigroups generated by affine functions, namely, . To prove our statement, it suffices to prove that every element of the form T(t)x where ∈R\Q belongs to the closure of {T(q)x:q ∈Q} in B. Now we prove it following lemma , which gives a sharp norm estimate of the operator eξL for all ξ ∈C.
It has already been mentioned in the introduction that the chaotic dynamics of the heat semigroup generated by certain shifts of the Laplace-Beltrami operator are extensively studied on symmetric spaces [21, 31] and harmonic NA groups [36]. The geometric interpretation of the above result can also be seen in the following figure. Thus, given the weak-type inequality (2,2) constrained above, it is natural to investigate whether the same holds for an operator given by right-hand convolution by a larger kernel, namely φ(x).
The presence of the oscillating factor, that is, Re(c(z)qiz) inφzi, is mainly responsible for the limited inequality of the weak type (2,2).
