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A theorem of Strichartz for multipliers on homogeneous trees

Sumit Kumar Rano, Rudra P. Sarkar

Abstract

A theorem of Strichartz states that if a uniformly bounded bi-infinite sequence of functions on the Euclidean spaces, satisfies the condition that the Laplacian acting on a function in this sequence yields the next one, then each function in this sequence is an eigenfunction of the Laplacian. We consider a generalization of this result for homogeneous trees, where we replace bounded functions by tempered distributions and the Laplacian by multiplier operators acting on the tempered distributions. After establishing the result in this general context, we narrow our focus to specific cases, which includes important examples of multiplier operators such as heat and Schrödinger operator, ball and sphere averages of function.

A theorem of Strichartz for multipliers on homogeneous trees

Abstract

A theorem of Strichartz states that if a uniformly bounded bi-infinite sequence of functions on the Euclidean spaces, satisfies the condition that the Laplacian acting on a function in this sequence yields the next one, then each function in this sequence is an eigenfunction of the Laplacian. We consider a generalization of this result for homogeneous trees, where we replace bounded functions by tempered distributions and the Laplacian by multiplier operators acting on the tempered distributions. After establishing the result in this general context, we narrow our focus to specific cases, which includes important examples of multiplier operators such as heat and Schrödinger operator, ball and sphere averages of function.
Paper Structure (17 sections, 22 theorems, 149 equations)

This paper contains 17 sections, 22 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalities
  4. Weak lp spaces
  5. Homogeneous trees
  6. The Laplacian and the Poisson transform
  7. Fourier transforms on fouriertransformonx
  8. Schwartz spaces on schwartzspacesonx
  9. Tempered distributions on tempereddistributionsonx
  10. Characterization of Eigenfunctions as Poisson Transform
  11. Multiplier Operators on Schwartz Spaces
  12. Strichartz's theorem for multipliers
  13. Strichartz's theorem for a restricted subclass of multipliers
  14. The Laplacian as a multiplier
  15. Averaging operators
  16. ...and 2 more sections

Key Result

Theorem 1.1

Suppose that a bi-infinite sequence $\{f_{k}\}_{k\in\mathbb{Z}}$ of functions on $\mathbb{R}^{n}$ satisfies $\Delta_{\mathbb{R}^{n}} f_{k}=f_{k+1}$ and $\|f_{k}\|_{L^{\infty}(\mathbb{R}^{n})}\leq M$, for all $k\in\mathbb{Z}$ and for a constant $M>0$. Then $\Delta_{\mathbb{R}^{n}} f_{0}=-f_{0}$.

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 4.1
  • ...and 30 more