Eigenfunctions of the Laplacian Satisfying Hardy-Type Estimates on Homogeneous Trees

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