Pseudo-Differential Calculus on Homogeneous Trees

To study concentration and oscillation properties of eigenfunctions of the discrete Laplacian on regular graphs, we construct in this paper a pseudo-differential calculus on homogeneous trees, their universal covers. We define symbol classes and associated operators. We prove that these operators are bounded on L (2) and give adjoint and product formulas. Finally, we compute the symbol of the commutator of a pseudo-differential operator with the Laplacian.