Symmetry breaking for representations of rank one orthogonal groups

We give a complete classification of intertwining operators (symmetry breaking operators) between spherical principal series representations of G=O(n+1,1) and G'=O(n,1). We construct three meromorphic families of the symmetry breaking operators, and find their distribution kernels and their residues at all poles explicitly.Symmetry breaking operators at exceptional discrete parameters are thoroughly studied. We obtain closed formulae for the functional equations which the composition of the the symmetry breaking operators with the Knapp-Stein intertwining operators of $G$ and G' satisfy, and use them to determine the symmetry breaking operators between irreducible composition factors of the spherical principal series representations of G and G'. Some applications are included.