$L^p$-asymptotic behaviour of approximate identities on homogeneous groups

Let $G$ be a homogeneous group. We show that $$\lim_{t \to \infty}\|\psi_t\|_{L^p(G)}^{-1}\|\mu \ast \psi_t-\mu (G)\psi_t\|_{L^p(G)}= 0$$ for $p\in [1, \infty ]$, where $\mu $ is any complex Borel measure on $G$, and $\{\psi _t : t \gt 0\}$ is a suitable approximate identity on $G$. The above result is a generalization of a well-known result which states that solutions of the heat equation on $\mathbb R^n$ with $L^1$ initial data behave asymptotically as the mass times the fundamental solution. We apply our result to study the asymptotic behaviour of solutions to certain initial value problems (e.g., fractional heat equation, extension problem etc.) on various spaces.