Feeling the heat in a group of Heisenberg type

In this paper we use the heat equation in a group of Heisenberg type G to provide unified treatment two very different extension problems for time independent pseudo-differential operators Ls and Ls, 0<s≤1. Here, is fractional power horizontal Laplacian, conformal Laplacian on G. One our main objective compute explicitly fundamental solutions these nonlocal by new approach exclusively based partial differential equations semigroup methods. When s=1 results recapture famous solution found Folland generalised Kaplan.