Fundamental solutions for the two dimensional affine group and Hn+1Hn+1

We derive the wave and heat kernels on the ax+bax+b group, as well as the fundamental solution of the group Laplacian. We make particular use of the Kontorovich–Lebedev transform and a recent result of the author to produce new expressions for these kernels. Our results easily extend to the hyperbolic space Hn+1Hn+1 for any n and the explicit formulas are given in n dimensions.