| Harmonic analysis on trees has been extensively studied in the last ten years. Many of results has been established on trees; for example, Hardy Spaces and their characterizations, Poisson integral representations, Hilbert transforms. Most of these results focus on the boundary of the tree and have a close connection with martingale techniques. Later, R. Rochberg and M. Taibleson studied the tree in a different way by considering the interior of the tree as the basic domain rather than the boundary. They gave factorizations of the Laplacian and the Green's operator and proved that the Laplacian is the inverse of the Green's operator on {dollar}lsp{lcub}p{rcub}, 0 < p < infty{dollar}, and that there is a {dollar}psb0, psb0geq 1{dollar}, such that the Green's operator is of strong type (p, p) for {dollar}psb0 < p < infty{dollar}. They also proved that if the random walk is strongly reversible then the Green's operator is of weak type (1,1). They suggested that we can define Hardy Space {dollar}Hsp1{dollar} in terms of Green's operator and study the dual space of {dollar}Hsp1{dollar} space. This thesis continues this line of development. We study the Hardy space {dollar}Hsp1{dollar} and its dual space BMO, the Green's operator G and its conjugate operator G* on {dollar}lspinfty{dollar}, an averaging operator, and the Marcinkiewicz type interpolation theory. |
