From the book "$C_0$-semigroups and Applications" of Vrabie, we have this defintion of uniformly bounded analytic semigroups:
Definition 7.1.1. Let $\mathbb{C}_\theta=\{z \in \mathbb{C} ;-\theta<\arg z<\theta\}$.
We say that the $C_0$-semigroup $\{S(t) ; t \geq 0\}$ is analytic, if there exists $0<\theta \leq \pi$, and a mapping $\tilde{S}: \overline{\mathbb{C}}_\theta \rightarrow \mathcal{L}(X)$ such that:
(i) $S(t)=\tilde{S}(t)$ for each $t \geq 0$; (ii) $\tilde{S}(z+w)=\tilde{S}(z) \tilde{S}(w)$ for each $z, w \in \overline{\mathbb{C}}_\theta$;
(iii) $\lim _{z \in \overline{\mathbb{C}}_\theta, z \rightarrow 0} \tilde{S}(z) x=x$ for each $x \in X$; (iv) the mapping $z \mapsto \tilde{S}(z)$ is analytic from $\mathbb{C}_\theta$ to $\mathcal{L}(X)$.
If, in addition, for each $0<\delta<\theta$, the mapping $z \mapsto \tilde{S}(z)$ is bounded from $\mathbb{C}_\delta$ to $\mathcal{L}(X)$, the $C_0$-semigroup $\{S(t) ; t \geq 0\}$ is called analytic and uniformly bounded.
Thence, if I'm not mistaken, uniformly bounded means bounded on all the subsectors. However, in many papers, the authors talk about "bounded analytic semigroups" without the term "uniformly". Is it a matter of terminology?
Thank you very much in advance for your help.
Yes, in the littrature, many authors use different termenology to mean the same thing: bounded analytic, bounded holomorphic, uniformly bounded analytic, uniformly bounded holomorphic,..
Note that a semigroup might be (uniformly) bounded on the real axis but not on a sector.
To be sure, I recommend to refer to the definition given in the reference in consideration.