Wiener-Hopf theory and nonunimodular groups
Let \(G\) be a connected real nonunimodular (i.e. the modular function \(m(g) \not\equiv 1)\) Lie group. Let \(X_0, \dots, X_k\) be the left invariant fields on \(G\), let \(\Delta = \sum_{j \geq 1} X^2_j + X_0\) and \(T_t = e^{-t \Delta}\) be the left Laplacian and the left diffusion semigroup, respectively. Let \(p_t(x,y) = \phi_t (y^{-1} x)\) \((x, y \in G)\) be the corresponding ``left'' diffusion kernel. The main result of the paper is \[ \phi_t (g) = O(t^{-3/2}) \quad \forall g \in G, \quad t \to \infty. \]
scientific article; zbMATH DE number 599772