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, $\Phi_t:\Omega\to\Omega$ 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 $$\Phi_t(z)\to z\,,\quad t\to0^+,$$ 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 $\mathbb D$. 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 $$\sup_{z\in\mathbb D}|\Phi_t(z)-z|= O(\sqrt t), \ t\to0^+.$$ 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 the right half-plane $\mathbb C_+$, we consider semigroups $\{\Phi_t\}$ with $\infty$ as its Denjoy-Wolff point. It is not difficult to show that Gumenyuk's result is no longer true for these semigroups. Our second result characterises when such continuous semigroups converge uniformly to the identity, as $t\to0^+$, in terms of their infinitesimal generators. Namely, this convergence holds if and only if the infinitesimal generator of the semigroup is bounded in the half-plane $\{z\in \mathbb C:\, \Re z>1\}$. In this case, we can also prove that the rate of convergence is again $O(\sqrt{t})$, as $t\to0^+$. An example of application of this result is when the semigroup is in the Gordon-Hedenmalm class (the one producing bounded composition operators on Hardy spaces of Dirichlet series). An important ingredient in the proofs of these results is harmonic measure, which we have done through a classic result of M. Lavrentiev.