A characterization of weak Lp-eigenfunctions of the Laplacian on homogeneous trees

Abstract

This work deals with the characterization of eigenfunctions of the Laplacian L on a homogeneous tree X, which satisfy certain growth conditions. More precisely, we shall prove that the Poisson transform on X provides an one-to-one correspondence between the subspace of all Hardy-type eigenfunctions of L on X and the Lebesgue spaces (possibly the set of all complex measures) on the boundary of X.

Dynamics of semigroups generated by analytic functions of the Laplacian on Homogeneous Trees

“…Using the estimate of Poisson transform (see Proposition 2.2) and the above duality relation, the authors of this paper proved the following (see [11]):…”

“…We now enlist some important properties of φ z in the following lemma, most of which follows easily from the explicit formula above (for details see [7,11]).…”

“…We also need the following estimates of P z F , which can be considered as a generalisation of the size estimates of φ z given above. For details about this topic we refer [11].…”

Dynamics of semigroups generated by analytic functions of the Laplacian on Homogeneous Trees

“…The estimates of the Poisson transform from Proposition 3 and the duality relation ( 5) together gives us the following 'restriction type' inequality for the Helgason-Fourier transform (see [10,Theorem 4.2] for a more general version):…”

A Result of Paley and Wiener on Damek–ricci spaces

“…To the best of our knowledge, the only related work for p = 2 is due to Kumar [30] on the rank one symmetric spaces of noncompact type. The proof given in [30] can be easily extended to harmonic N A groups (see also [31,Theorem 4.3]). In [30], the author proved that P λ is restricted weak type (2, 2) but without any dependence of the norm of P λ with the parameter λ.…”