Eigenfunctions of the Laplacian satisfying Hardy-type estimates on homogeneous trees

Abstract

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