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

Abstract

In this article we prove the restriction theorem for Helgason-Fourier transform on homogeneous tree. Our proof is based on the duality argument and the norm estimates of Poisson transform. We also characterize all eigenfunctions of the laplacian on homogeneous tree which are Poisson transform of L p functions defined on the boundary.

Pointwise convergence of the heat and subordinates of the heat semigroups associated with the Laplace operator on homogeneous trees and two weighted $L^p$ maximal inequalities

“…In the last years the study of harmonic analysis in the homogeneous trees has turned to take great interest. Heat and Poisson semigroups ( [24] and [31]), Poincaré and Hardy inequalities [5], Hardy and BMO spaces ( [3,4,7]), nondoubling flow measures ( [25,26]), uncertainty principles ( [15]), Carleson measures ( [12]), special multipliers ( [8]) and maximal functions ( [14,18,28,29]) are some of the topics that have being recently studied in this setting.…”

A Theorem of Roe and Strichartz on homogeneous trees

“…In [12] it was proved that the theorem indeed remains valid when uniform boundedness is replaced by uniform "almost L p boundedness". Here it is worth mentioning that these size estimates arise naturally due to the behaviour of the Poisson transforms, which also acts as the eigenfunctions of L with the eigenvalues lying in the interior of the ellipse (2.5) (For details see [11]). The version of Roe's theorem that we have proved in this article in the context of homogeneous trees are the following.…”

“…It was proved in [11] that any weak L p eigenfunction of the Laplacian of X can be represented by the Poisson transform of a L p function on the boundary. Therefore the conclusion of these theorems is more precise than that of Theorem 1.1.…”