H(z)=F_\mu^{-1}(z)=pz+(1-p)F_{\mu_p}(z),\;\;\;\;\;z\in\mathbb{C}^+,$$ $\phi_\mu=E_{\mathbb{B}^{-1}(\mu)}$, and $$F_{\mu^{\uplus
p^*}}(z)=F_{\mu_p}(F_\mu(z)),\;\;\;\;\;z\in\mathbb{C}^+,$$ where $$\label{mup} \mu_p=\left(\mu^{\uplus
p^*}\right)^{\boxplus\frac{1}{p}},\;\;\;\;\;p>1.$$ Moreover, for $r>0$ the measure $\mu^{\boxplus r}$ is $\boxplus$-infinitely divisible and $$\phi_{\mu^{\boxplus
r}}=E_{\left(\mu^{\uplus(1+r)}\right)^{\boxplus\frac{r}{1+r}}}.$$

First suppose that (2) holds, i.e., there exists some $\nu\in\mathcal{M}$ such that $H(z)=2z-F_\nu(z)$ and $H(F_\mu(z))=z$, $z\in\mathbb{C}^+$. By Proposition \[prop2.1\](5) we see that $F_\mu(z)=\left[F_{\nu^{\boxplus2}}(z)+z\right]/2$ or, equivalently, $\mu=\mathbb{B}_1(\nu)$, whence (1) holds by Proposition \[3.1\]. Conversely, if $\mu$ is $\boxplus$-infinitely divisible then the measure $\mu_p$ in (\[mup\]) is defined by Lemma \[3.4\] and Proposition \[3.5\]. Moreover, by the fact $\mathbb{B}_{p,1/p^*}(\mu_p)=\mu$ and Proposition \[3.1\], we obtain $\phi_\mu=E_{\mu_p^{\uplus(p-1)}}$, which yields the implications that (1) implies (2), and (\[H\]). The last assertion follows from (\[bijection\]) and Proposition \[3.6\](1). Indeed, we have $$\phi_{\mu^{\boxplus r}}=rE_{\mathbb{B}^{-1}(\mu)}=E_{((\mu^{\uplus2})^{\boxplus1/2})^{\uplus r}}
=E_{(\mu^{\uplus(1+r)})^{\boxplus r/(1+r)}},$$ as desired. This finishes the proof.

Next, we analyze the supports and regularity of $\boxplus$-infinitely divisible measures. Given such a measure $\mu$, let $\Omega=F_\mu(\mathbb{C}^+)$. Then by Proposition \[prop2.1\](1) and \[sub\], $\Omega=\{z\in\mathbb{C}^+:\Im
F_\mu^{-1}(z)>0\}$ is a simply connected domain and $\partial\Omega=F_\mu(\mathbb{R})$ is the graph of the continuous function $$\begin{aligned}
f(x)&=\inf\left\{y>0:\Im F_\mu^{-1}(x+iy)>0\right\} \\
&=\inf\left\{y>0:\frac{\Im
E_{\mathbb{B}^{-1}(\mu)}(x+iy)}{y}>-1\right\},\;\;\;\;\;x\in\mathbb{R}.\end{aligned}$$ Then Theorem \[Hthm\](1) shows that the function $\psi(x)=F_\mu^{-1}(x+if(x))$, $x\in\mathbb{R}$, is homeomorphism on $\mathbb{R}$. With the help of Proposition \[prop2.1\], we have the following conclusions.

\[3.10\] Suppose that $\mu\in\mathcal{M}$ is $\boxplus$-infinitely divisible.

1.  [The function $\phi_\mu$ has 