0, \\
z_{1}=r_{1}e^{i\varphi _{1}},\dots ,z_{n}=r_{n}e^{i\varphi _{n}}\in D,\quad
z_{1},\dots ,z_{n}\not=0, \\
u(0,z_{1},\dots ,z_{n})=f(z_{1},\dots ,z_{n}).
\end{gather*}
\end{remark}
\begin{remark} \label{rmk2.4} \rm
Reasoning as in the proof of Corollary \ref{cor1} and calculating
\begin{equation*}
\Big( \frac{d}{dt}H_{t}(f)(z_{1},\dots ,z_{n})\Big)\Big| _{t=0},
\end{equation*}
we easily find that the initial value problem
\begin{gather*}
\frac{\partial u}{\partial t}+\frac{1}{2}\sum_{k=1}^{n}\Big( z_{k}\frac{
\partial u}{\partial z_{k}}+z_{k}^{2}\frac{\partial ^{2}u}{\partial z_{k}^{2}
}\Big) =0,\quad (t,z_{1},\dots ,z_{n})\in \mathbb{R}_{+}\times (D\backslash
\{0\})^{n}, \\
u(0,z_{1},\dots ,z_{n})=f(z_{1},\dots ,z_{n}),\quad
z_{1},\dots ,z_{n}\in D,
\end{gather*}
is well-posed and its unique solution is $H_{t}(f)\in C^{\infty }(\mathbb{R}
_{+};B_{\alpha }^p(D^{n}))$.
\end{remark}
\section{Laplace-type equations with complex spatial variables}
The first main result of this section is concerned with the Laplace equation
of a complex spatial variable.
\begin{theorem} \label{theo-26}
Let $1\leq p<+\infty $ and consider $Q_{t}(f)(z)$
given by \eqref{laplace}, for $z\in D$. Then $(Q_{t},t\geq 0)$ is a ($C_{0}$
)-contraction semigroup of linear operators on $B_{\alpha }^p(D)$ and the
unique solution $u(t,z)$ that belongs to $B_{\alpha }^p(D)$ for each fixed
$t$, of the Cauchy problem \eqref{equ2} with the initial condition
\begin{equation*}
u(0,z)=f(z),\quad z\in D,\; f\in B_{\alpha }^p(D),
\end{equation*}
is given by $u(t,z)=Q_{t}(f)(z)$.
\end{theorem}
\begin{proof}
By \cite[Theorem 3.1]{GGG1} (see also \cite[Theorem 2.3.1, p. 27]{GGG3}),
$Q_{t}(f)(z)$ is analytic in $D$ and for all $z\in D$, $t,s\geq 0$ we have
$Q_{t}(f)(z)=\sum_{k=0}^{\infty }a_{k}e^{-kt}z^{k}$ and
$Q_{t+s}(f)(z)=Q_{t}[Q_{s}(f)](z)$. Now, since
$\frac{t}{\pi }\int_{-\infty }^{+\infty }\frac{1}{u^{2}+t^{2}}du=1$, by
Jensen's inequality, we get
\[
|Q_{t}(f)(z)|^p\leq \frac{t}{\pi }\int_{-\infty }^{+\infty
}|f(ze^{-iu})|^p\frac{1}{u^{2}+t^{2}}du,
\]
which multiplied on both sides by $\rho _{\alpha }(z)$, then integrated on
$D $ with respect to the Lebesgue's area measure $dA(z)$ and applying the
Fubini's theorem, gives
\begin{equation*}
\int_{D}|Q_{t}(f)(z)|^pdA_{\alpha }(z)
\leq \frac{t}{\pi }\int_{-\infty}^{+\infty }
\Big[ \int_{D}|f(ze^{-iu})|^pdA_{\alpha }(z)\Big]
\frac{1}{u^{2}+t^{2}}du.
\end{equation*}
As in the proof of Theorem \ref{theo-26}, writing $z=re^{i\theta }$
(in polar coordinates) and taking into account that $dA(z)=\frac{1}{\pi }
rdrd\theta $, some simple calculations lead to the same equality
\eqref{again}. Hence, we get $\| Q_{t}(f)\| _{p,\alpha }\leq \| f\|
_{p,\alpha }$. This implies that $Q_{t}(f)\in B_{\alpha }^p(D)$ and that
$Q_{t}$ is a contraction.
To prove that $\lim_{t\searrow 0}Q_{t}(f)(z)=f(z)$, for any $f\in
B_{\alpha }^p(D)$ and $z\in D$, let $f=U+iV$, $z=re^{ix}$ be fixed with
$0