chapter  19
4 Pages

Convolutions on the Heisenberg Group

We are interested in solving for u of the initial value problem for the heat equation governed by the sub-Laplacian L, i.e.,{

∂u ∂ρ (z, t, ρ) = −(Lu)(z, t, ρ), u(z, t, 0) = f(z, t)

for all (z, t) ∈ H1 and all time ρ > 0. Formally, u is given by u(z, t, ρ) = (e−ρLf)(z, t), (z, t) ∈ H1, ρ > 0.