On the uniform convergence of continuous semigroups

Let $\Omega$ be a region in the complex plane $\mathbb C$ and let $\{\Phi_t \}_{t\ge 0}$ be a continuous semigroup of functions on $\Omega$; that is, $\Phi_t\colon \Omega\to\Omega$ is holomorphic for every $t\ge 0$, $\Phi_0(z)=z$, for every $z\in\Omega$, $\Phi_t\circ\Phi_s=\Phi_{s+t}$, for every $s$, $t\ge 0$, and \begin{equation*}\label{eso} \Phi_t(z)\to z\,,\quad\hbox{ as $t$ goes to $0^+$,} \end{equation*} uniformly on compact subsets of $\Omega$. Despite this definition only requires the uniform convergence on compact subsets, P. Gumenyuk proved in 2014 that, when $\Omega$ is the unit disc, the convergence is uniform on the whole $\mathbb D$. When $\Omega$ is a half-plane, it is not difficult to show that the result is no longer true. In this paper, we enhance Gumenyuk's result by proving that for every continuous semigroup $\{\Phi_t \}_{t\ge 0}$ on $\mathbb D$ we have $$ \sup_{z\in\mathbb D} |\Phi_t(z)-z|= O(\sqrt t), \quad t\to 0^+. $$ In addition, we provide an example showing that $O(\sqrt t)$ is the best possible rate of uniform convergence valid for all semigroups on $\mathbb D$. When $\Omega$ is a half-plane we prove that there is uniform convergence to the identity under certain boundedness conditions on the infinitesimal generator of the semigroup. These boundedness conditions are fulfilled when the semigroup $\{\Phi_t \}_{t\ge 0}$ is included in the Gordon-Hedenmalm class (the one which produces bounded composition operators on Hardy spaces of Dirichlet series). An important ingredient in the proofs of these results is the use of harmonic measures, which we have done through a classic result of M. Lavrentiev.