\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 236, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/236\hfil 
 Heat and Laplace equations in complex variables]
{Heat and Laplace type equations with complex spatial variables in weighted
Bergman spaces}

\author[C. G. Gal, S. G. Gal \hfil EJDE-2017/236\hfilneg]
{Ciprian G. Gal, Sorin G. Gal}

\address{Ciprian G. Gal \newline
Department of Mathematics,
Florida International University, Miami,
FL 33199, USA}
\email{cgal@fiu.edu} 

\address{Sorin G. Gal \newline
 University of Oradea,
Department of Mathematics and Computer Science,
Str. Universitatii Nr. 1, 410087 Oradea, Romania}
\email{galso@uoradea.ro}

\dedicatory{Communicated by Jerome A. Goldstein}

\thanks{Submitted August 25, 2017. Published September 29, 2017.}
\subjclass[2010]{47D03, 47D06, 47D60}
\keywords{Complex spatial variable; semigroups of linear operators;
\hfill\break\indent  heat equation; Laplace equation; weighted Bergman space}

\begin{abstract}
 In a recent book, the authors of this paper have studied the classical heat
 and Laplace equations with real time variable and complex spatial variable
 by the semigroup theory methods, under the hypothesis that the boundary
 function belongs to the space of analytic functions in the open unit disk
 and continuous in the closed unit disk, endowed with the uniform norm. The
 purpose of the present note is to show that the semigroup theory methods
 works for these evolution equations of complex spatial variables, under the
 hypothesis that the boundary function belongs to the much larger weighted
 Bergman space $B_{\alpha }^p(D)$ with $1\leq p<+\infty $, endowed with a
 $L^p$-norm. Also, the case of several complex variables is considered. The
 proofs require some new changes appealing to Jensen's inequality, Fubini's
 theorem for integrals and the $L^p$-integral modulus of continuity. The
 results obtained can be considered as complex analogues of those for the
 classical heat and Laplace equations in $L^p(\mathbb{R})$ spaces.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks


\section{Introduction}

Extending the method of semigroups of operators in solving the evolution
equations of real spatial variable, a way of ``complexifying'' the spatial
variable in the classical evolution equations is to ``complexify'' their
solution semigroups of operators, as it was summarized in the book 
\cite{GGG3}. In the cases of heat and Laplace equations and their higher order
correspondents, the results obtained can be summarized as follows.

Let $D=\{z\in \mathbb{C};|z|<1\}$ be the open unit disk and 
$A(D)=\{f\colon \overline{D}\to \mathbb{C}$; $f$ is analytic on $D$,
continuous on $\overline{D}\}$, endowed with the uniform norm 
$\|f\|=\sup \{|f(z)|:z\in \overline{D}\}$. It is well-known that 
$(A(D),\|\cdot \|)$ is a Banach space. Let $f\in A(D)$ and consider the operator
\begin{equation}
W_{t}(f)(z)=\frac{1}{\sqrt{2\pi t}}\int_{-\infty }^{+\infty
}f(ze^{-iu})e^{-u^{2}/(2t)}du,\quad z\in \overline{D}.  \label{heat}
\end{equation}
In \cite{GGG1} (see also \cite[Chapter 2]{GGG3}, for more details) it was
proved that $(W_{t},t\geq 0)$ is a ($C_{0}$)-contraction semigroup of linear
operators on $A(D)$ and that the unique solution $u(t,z)$ (that belongs to 
$A(D)$, for each fixed $t\geq 0$) of the Cauchy problem
\begin{gather}
\frac{\partial u}{\partial t}(t,z)=\frac{1}{2}\frac{{\partial }^{2}u}{
\partial {\varphi }^{2}}(t,z),\quad (t,z)\in ( 0,+\infty )
\times D,\; z=re^{i\varphi },\; z\not=0,  \label{equ1} \\
u(0,z)=f(z),\quad z\in \overline{D},\; f\in A(D),  \label{bound1}
\end{gather}
is exactly
\begin{equation}
u(t,z)=W_{t}(f)(z).  \label{sol1}
\end{equation}
In the same contribution \cite{GGG1}, setting
\begin{equation}
Q_{t}(f)(z):=\frac{t}{\pi }\int_{-\infty }^{+\infty }\frac{f(ze^{-iu})}{
u^{2}+t^{2}}\,du,\quad z\in \overline{D},  \label{laplace}
\end{equation}
we proved that $(Q_{t},t\geq 0)$ is a ($C_{0}$)-contraction semigroup of
linear operators on $A(D)$. Consequently, the unique solution $u(t,z)$ (that
belongs to $A(D)$, for each fixed $\mathit{t}$) of the Cauchy problem
\begin{gather}
\frac{{\partial }^{2}u}{\partial t^{2}}(t,z)+\frac{{\partial }^{2}u}{
\partial {\varphi }^{2}}(t,z)=0,\quad (t,z)\in D\times ( 0,+\infty
) ,\; z=re^{i\varphi },\; z\not=0,  \label{equ2} \\
u(0,z)=f(z),\quad z\in \overline{D},\quad f\in A(D),  \label{bound2}
\end{gather}
is exactly
\begin{equation}
u(t,z)=Q_{t}(f)(z).  \label{sol2}
\end{equation}

The goal of the present note is to show that the well-posedness of the above
problems in the space $A(D)$, can be replaced by well-posedness in some
(larger) weighted Bergman spaces defined in what follows. 
For $0<p<+\infty $, $1<\alpha <+\infty $ and 
$\rho _{\alpha }(z)=(\alpha+1)(1-|z|^{2})^{\alpha }$, the weighted 
Bergman space $B_{\alpha }^p(D)$,
is the space of all analytic functions in $D$, such that
\begin{equation*}
\Big[ \int_{D}|f(z)|^pdA_{\alpha }(z)\Big] ^{1/p}<+\infty ,
\end{equation*}
where $dA_{\alpha }(z)=\rho _{\alpha }(z)dA(z)$, with
\begin{equation*}
dA(z)=\frac{1}{\pi }dxdy=\frac{1}{\pi }rdrd\theta ,\quad
z=x+iy=re^{i\theta },
\end{equation*}
the normalized Lebesgue area measure on the unit disk of the complex plane.
Setting
\begin{equation*}
\| f\| _{p,\alpha }=( \int_{D}|f(z)|^pdA_{\alpha }(z))^{1/p},
\end{equation*}
for $1\leq p<+\infty$, it is well-known that 
$(B_{\alpha }^p(D),\| \cdot \| _{p,\alpha })$ is a Banach space,
 while for $0<p<1$, $B_{\alpha}^p(D)$ is a complete metric space with
metric $d(f,g)=\| f-g\|_{p,\alpha }^p$. For other details concerning
Bergman spaces in the complex plane and their properties, we refer the 
reader to the books \cite[Chapter 1, Section 1.1]{Heden} and 
\cite[pp. 30-32]{Duren}.

The results obtained can be considered as complex analogues of those for the
classical heat and Laplace equations in $L^p(\mathbb{R})$ spaces
(see, e.g., \cite[p. 23]{Gold1}). In Section 2, we reconsider \eqref{heat},
\eqref{equ1}, \eqref{bound1}, \eqref{sol1} assuming that the boundary function 
$f\in B_{\alpha }^p(D)$ with $1\leq p<+\infty $. Section 3 treats
\eqref{laplace}, \eqref{equ2}, \eqref{bound2}, \eqref{sol2} under the same
hypothesis for the boundary function $f$. It is worth mentioning that since
the uniform norm used in the case of the space $A(D)$ is now replaced with
the $L^p$-type norm in the Bergman space $B_{\alpha }^p(D)$, the proofs
of these results require now some changes based on new tools, like the
Jensen's inequality, the Fubini's theorem for integrals and the $L^p$
-integral modulus of continuity.

\section{Heat-type equations with complex spatial variables}

The first main result of this section is concerned with the heat 
equation of complex spatial variable.

\begin{theorem} \label{theo-25}
Let $1\leq p<+\infty $ and consider $W_{t}(f)(z)$
given by \eqref{heat}, for $z\in D$. Then, $(W_{t},t\geq 0)$ 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{equ1} 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 $u(t,z)=W_{t}(f)(z)$.
\end{theorem}

\begin{proof}
By \cite[Theorem 2.1]{GGG1} (see also \cite[Theorem 2.2.1, p. 27]{GGG3}), 
$W_{t}(f)(z)$ is analytic in $D$ and for all $z\in D$, $t,s\geq 0,$ we have 
$W_{t}(f)(z)=\sum_{k=0}^{\infty }a_{k}e^{-k^{2}t/2}z^{k}$ and 
$W_{t+s}(f)(z)=W_{t}[W_{s}(f)](z)$.

In what follows, we apply the following well-known Jensen type inequality
for integrals: if $\int_{-\infty }^{+\infty }G(u)du=1$, $G(u)\geq 0$ for all
$u\in \mathbb{R}$ and $\varphi (t)$ is a convex function over the range of
the measurable function of real variable $F$, then
\begin{equation*}
\varphi \Big( \int_{-\infty }^{+\infty }F(u)G(u)du\Big) 
\leq \int_{-\infty }^{+\infty }\varphi (F(u))G(u)du.
\end{equation*}
Now, since $\int_{-\infty }^{+\infty }\frac{1}{\sqrt{2\pi t}}
e^{-u^{2}/(2t)}du=1$, by the above Jensen's inequality with 
$\varphi (t)=t^p$, $F(u)=|f(ze^{-iu})|$ and
$G(u)=\frac{1}{\sqrt{2\pi t}}e^{-u^{2}/(2t)}du$, we find
\begin{equation*}
|W_{t}(f)(z)|^p\leq \frac{1}{\sqrt{2\pi t}}\int_{-\infty }^{+\infty
}|f(ze^{-iu})|^pe^{-u^{2}/(2t)}du.
\end{equation*}

Multiplying this inequality by 
$\rho _{\alpha }(z)=(\alpha +1)(1+|z|^{2})^{\alpha }$, integrating on $D$ 
with respect to $dA(z)$ (the normalized Lebesgue's area measure) and taking 
into account the Fubini's theorem, we obtain
\begin{equation*}
\int_{D}|W_{t}(f)(z)|^pdA_{\alpha }(z)\leq \frac{1}{\sqrt{2\pi t}}
\int_{-\infty }^{+\infty }\Big[ \int_{D}|f(ze^{-iu})|^pdA_{\alpha }(z)
\Big] e^{-u^{2}/(2t)}du.
\end{equation*}

But 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
equality
\begin{equation}
\int_{D}|f(ze^{-iu})|^pdA_{\alpha }(z)=\int_{D}|f(z)|^pdA_{\alpha }(z),
\mbox{ for all }u\in \mathbb{R}\text{.}  \label{again}
\end{equation}
This immediately implies
\begin{equation*}
\| W_{t}(f)\| _{p,\alpha }\leq \| f\| _{p,\alpha }.
\end{equation*}
So $W_{t}(f)\in B_{\alpha }^p(D)$ and $W_{t}$ is a contraction. Next, for
$f\in B_{\alpha }^p(D)$ let us introduce the integral modulus of continuity
(see, e.g., \cite{Blas})
\begin{equation*}
\omega _{1}(f;\delta )_{B_{\alpha }^p}=\sup_{0\leq |h|\leq \delta }\Big(
\int_{D}|f(ze^{ih})-f(z)|^pdA_{\alpha }(z)\Big) ^{1/p}.
\end{equation*}

Reasoning as above and taking into account that $\int_{D}1dA(z)=\frac{1}{\pi
}\int_{D}dxdy=1$, we obtain
\begin{align*}
& \int_{D}|W_{t}(f)(z)-f(z)|^pdA_{\alpha }(z) \\
& \leq \frac{1}{\sqrt{2\pi t}}\int_{-\infty }^{+\infty }\Big[
\int_{D}|f(ze^{-iu})-f(z)|^pdA_{\alpha }(z)\Big] e^{-u^{2}/(2t)}du \\
& \leq \frac{1}{\sqrt{2\pi t}}\int_{-\infty }^{+\infty }\omega
_{1}(f;|u|)_{B_{\alpha }^p}^pe^{-u^{2}/(2t)}\,du \\
& \leq \frac{1}{\sqrt{2\pi t}}\int_{-\infty }^{+\infty }\omega _{1}(f;\sqrt{t
})_{B_{\alpha }^p}^p\Big( \frac{|u|}{\sqrt{t}}+1\Big)^pe^{-u^{2}/(2t)}\,du \\
& =2\frac{\sqrt{t}}{\sqrt{2\pi t}}\cdot \int_{0}^{\infty }\omega _{1}
(f;\sqrt{t})_{B_{\alpha }^p}^p(v+1)^pe^{-v^{2}/2}dv \\
&=C_{p}\omega _{1}(f; \sqrt{t})_{B_{\alpha }^p}^p.
\end{align*}
Consequently, we get
\begin{equation*}
\| W_{t}(f)-f\| _{B_{\alpha }^p}\leq C_{p}^{\prime }\omega _{1}(f;
\sqrt{t})_{B_{\alpha }^p},
\end{equation*}
and so it yields that $\lim_{t\searrow 0}\| W_{t}(f)-f\| _{B_{\alpha
}^p}=0$. Since it is well-known that (see, e.g., \cite[pg. 3, proof of
Proposition 1.2]{Heden}) the convergence in the Bergman space means uniform
convergence in any compact subset of $D$, it follows that $\lim_{t\searrow
0}W_{t}(f)(z)=f(z)$, for all $z\in D$.

Now, let $s\in (0,+\infty )$, $V_{s}$ be a small neighborhood of $s$, both
fixed, and take an arbitrary $t\in V_{s}$, $t\not=s$. Applying the
reasonings in the proof of \cite[Theorem 2.1, (iii)]{GGG1} 
(see also \cite[Theorem 2.2.1, (iii)]{GGG3}), we get
\begin{align*}
|W_{t}(f)(z)-W_{s}(f)(z)|
& \leq \int_{-\infty }^{+\infty
}|f(ze^{-iu})|\big| \frac{e^{-u^{2}/(2t)}}{\sqrt{2\pi t}}-\frac{
e^{-u^{2}/(2s)}}{\sqrt{2\pi s}}\big| du \\
& \leq \frac{1}{\sqrt{2\pi }}|t-s|\int_{-\infty }^{+\infty
}|f(ze^{-iu})|e^{-u^{2}/c^{2}}\big| \frac{2u^{2}}{c^{4}}-\frac{1}{c^{2}}
\big| du,
\end{align*}
where $c$ depends on $s$ (and not on $t$).

Denoting $K_{s}=\int_{-\infty }^{+\infty }e^{-u^{2}/c^{2}}| \frac{
2u^{2}}{c^{4}}-\frac{1}{c^{2}}| du$, i.e., where $0<K_{s}<+\infty $, by the proof 
of \cite[Theorem 2.2.1, (iii)]{GGG3}, it follows that
\begin{align*}
&\int_{D}|W_{t}(f)(z)-W_{s}(f)(z)|^pdA_{\alpha }(z) \\
&\leq \big( \frac{1}{\sqrt{2\pi }}\big)^p
 |t-s|^pK_{s}^p\int_{D}\Big( \int_{-\infty }^{+\infty }|f(ze^{-iu})|
\frac{1}{K_{s}}e^{-u^{2}/c^{2}}\big| \frac{2u^{2}}{c^{4}}-\frac{1}{c^{2}
}\big| du\Big) ^pdA_{\alpha }(z)
\end{align*}
and reasoning exactly as at the beginning of the proof, we obtain
\begin{equation*}
\int_{D}|W_{t}(f)(z)-W_{s}(f)(z)|^pdA_{\alpha }(z)
\leq \big( \frac{1}{
\sqrt{2\pi }}\big) ^p|t-s|^pK_{s}^p\| f\| _{p,\alpha }^p.
\end{equation*}
This implies
\begin{equation*}
\| W_{t}(f)-W_{s}(f)\| _{p,\alpha }\leq \frac{1}{\sqrt{2\pi }}
|t-s|K_{s}\| f\| _{p,\alpha }.
\end{equation*}
Therefore, $(W_{t},t\geq 0)$ is a ($C_{0}$)-contraction semigroup of linear
operators on $B_{\alpha }^p(D)$. Also, since the series representation for
$W_{t}(f)(z)$ is uniformly convergent in any compact disk included in $D$,
it can be differentiated term by term with respect to $t$ and $\varphi $. We
then easily obtain that $W_{t}(f)(z)$ satisfies the Cauchy problem in the
statement of the theorem. We also note that in equation we must take $
z\not=0 $ simply because $z=0$ has no polar representation, namely $z=0$
cannot be represented as function of $\varphi $. This completes the proof.
\end{proof}

The partial differential equation \eqref{equ1} can equivalently be expressed
in terms of $t$ and $z$ as follows.

\begin{corollary}\label{cor1}
Let $1\leq p<+\infty $. For each $f\in B_{\alpha
}^p(D) $, the initial value problem
\begin{equation*}
\frac{\partial u}{\partial t}+\frac{1}{2}\Big( z\frac{\partial u}{\partial z
}+z^{2}\frac{\partial ^{2}u}{\partial z^{2}}\Big) =0,\quad 
(t,z)\in \mathbb{R}_{+}\times D\backslash \{0\},\;
 u(0,z)=f(z),z\in D, \end{equation*}
is well-posed and its unique solution is 
$W_{t}(f)\in C^{\infty }(\mathbb{R}_{+};B_{\alpha }^p(D))$.
\end{corollary}

\begin{proof}
By \cite[Theorem 2.1, (i)]{GGG1}, we can compute the generator associated
with $W_{t}(f)$, to get
\begin{align*}
\big( \frac{d}{dt}W_{t}(f)(z)\big) _{|t=0}
& =-\sum_{k=0}^{\infty }\frac{k^{2}}{2}a_{k}z^{k}
=-\sum_{k=0}^{\infty }( \frac{k(k-1)}{2}+\frac{k}{2}) a_{k}z^{k} \\
& =-\frac{z^{2}}{2}f^{\prime \prime }(z)-\frac{z}{2}f^{\prime }(z).
\end{align*}
Therefore, the statement is an immediate consequence of a classical result
of Hill (see, e.g., \cite[Theorem 1.2.1, pg. 8]{GGG3}).
\end{proof}

\begin{remark}\label{rmk2.3} \rm
Theorem \ref{theo-25} can be easily extended to functions of several complex
variables, as follows. For $0<p<\infty $, let 
$f:D^{n}\to \mathbb{C}$, $f(z_{1},z_{2},\dots ,z_{n})$
be analytic with respect to each variable $z_{i}\in D$, $i=1,\dots ,n$, such that
\begin{equation*}
\int_{D^{n}}|f(z_{1},\dots ,z_{n})|^pdA_{\alpha }(z_{1})\dots dA_{\alpha
}(z_{n})<\infty .
\end{equation*}
We write this by $f\in B_{\alpha }^p(D^{n})$ and
\begin{equation*}
\| f\| _{B_{\alpha }^p(D^{n})}
=\Big(\int_{D^{n}}|f(z_{1},\dots ,z_{n})|^pdA_{\alpha }(z_{1})\dots 
dA_{\alpha}(z_{n})\Big) ^{1/p}
\end{equation*}
becomes a norm on $B_{\alpha }^p(D^{n})$. We call the later the weighted
Bergman space in several complex variables. Following now the model for the
semigroup attached to the real multivariate heat equation (see, e.g., 
\cite[pg. 69]{Gold1}), for $z_{1},z_{2},\dots ,z_{n}\in D,$ we may define the
integral of Gauss-Weierstrass type by
\begin{equation*}
H_{t}(f)(z_{1},\dots ,z_{n})=(2\pi t)^{-n/2}
\int_{D^{n}}f(z_{1}e^{-iu_{1}},\dots ,z_{n}e^{-iu_{n}})
 e^{-|u|^{2}/(2t)}du_{1}\dots du_{n},
\end{equation*}
where $u=(u_{1},\dots ,u_{n})$ and $|u|=\sqrt{u_{1}^{2}+\dots +u_{n}^{2}}$. One
can reason exactly as in the proof of Theorem \ref{theo-25} to deduce that 
$(H_{t}$, $t\geq 0)$ is ($C_{0}$)-contraction semigroup of linear operators
on $B_{\alpha }^p(D^{n})$ and that
$u(t,z_{1},\dots ,z_{n})=H_{t}(f)(z_{1},\dots ,z_{n})$ is the unique solution of
the Cauchy problem
\begin{gather*}
\frac{\partial u}{\partial t}(t,z_{1},\dots ,z_{n})
=\frac{1}{2}\big[ \frac{{
\partial }^{2}u}{\partial {\varphi _{1}}^{2}}(t,z_{1},\dots ,z_{n})+\dots +\frac{{
\partial }^{2}u}{\partial {\varphi _{n}}^{2}}(t,z_{1},\dots ,z_{n})\big] ,
\quad t>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
$0<r\leq \rho <1$ and denote
\begin{equation*}
F(v)=U[r\cos(v),r\sin(v)],G(v)=V[r\cos(v),r\sin(v)].
\end{equation*}
We can write
\begin{equation*}
Q_{t}(f)(z)=\frac{t}{\pi }\int_{-\infty }^{+\infty }F(x-u)\frac{1}{
t^{2}+u^{2}}du+i\frac{t}{\pi }\int_{-\infty }^{+\infty }G(x-u)\frac{1}{
t^{2}+u^{2}}du.
\end{equation*}
From the maximum modulus principle, when estimating the quantity 
$|Q_{t}(f)(z)-f(z)|$, for $|z|\leq \rho <1$, we may take $r=\rho $. Now,
passing to limit as $t\to 0^{+}$ and taking into account the
property in the real case (see, e.g., \cite[p. 23, Exercise 2.18.8]{Gold1}),
we find
\begin{align*}
\lim_{t\searrow 0}|Q_{t}(f)(z)-f(z)|
& \leq \lim_{t\searrow 0}\Big|
\frac{t}{\pi }\int_{-\infty }^{+\infty }F(x-u)\frac{1}{t^{2}+u^{2}}
du-F(x)\Big| \\
&\quad +\lim_{t\searrow 0}\Big| \frac{t}{\pi }\int_{-\infty }^{+\infty
}G(x-u)\frac{1}{t^{2}+u^{2}}du-G(x)\Big| 
 =0,
\end{align*}
which holds uniformly with respect to $|z|\leq \rho $. Consequently, it
follows that $\lim_{t\searrow 0}Q_{t}(f)(z)=f(z)$, uniformly in any compact
subset of $D$.

Now, let $s\in (0,+\infty )$, $V_{s}$ be a small neighborhood of $s$, 
both fixed, and take an arbitrary $t\in V_{s}$, $t\not=s$. Applying the same
reasoning as in the proof of \cite[Theorem 3.1, (ii)]{GGG1} 
(see also \cite[Theorem 2.3.1, (ii)]{GGG3}), we get
\begin{align*}
|Q_{t}(f)(z)-Q_{s}(f)(z)|
& \leq \frac{1}{\pi }\int_{-\infty }^{+\infty }|f(ze^{-iu})|
\big| \frac{t}{t^{2}+u^{2}}-\frac{s}{s^{2}+u^{2}}
\big| du \\
& =\frac{1}{\pi }|t-s|\int_{-\infty }^{+\infty }|f(ze^{-iu})|\big|
\frac{u^{2}-ts}{(t^{2}+u^{2})(s^{2}+u^{2})}\big| du.
\end{align*}
Setting 
\[
K_{s}=\int_{-\infty }^{+\infty }\left| \frac{u^{2}-ts}{
(t^{2}+u^{2})(s^{2}+u^{2})}\right| du,
\]
 i.e.,
\begin{equation*}
1=\frac{1}{K_{s}}\int_{-\infty }^{+\infty }\big| \frac{u^{2}-ts}{
(t^{2}+u^{2})(s^{2}+u^{2})}\big| du
\end{equation*}
(where $0<K_{s}<+\infty $), by the proof of \cite[Theorem 2.3.1, (ii)]{GGG3}, 
it follows that
\begin{align*}
& \int_{D}|Q_{t}(f)(z)-Q_{s}(f)(z)|^pdA_{\alpha }(z) \\
& \leq \big( \frac{1}{\pi }\big) ^p|t-s|^pK_{s}^p\int_{D}
\Big(\int_{-\infty }^{+\infty }|f(ze^{-iu})|\frac{1}{K_{s}}\big| \frac{
u^{2}-ts}{(t^{2}+u^{2})(s^{2}+u^{2})}\big| du\Big) ^pdA_{\alpha }(z).
\end{align*}
Applying Jensen's inequality, we obtain
\begin{equation*}
\int_{D}|Q_{t}(f)(z)-Q_{s}(f)(z)|^pdA_{\alpha }(z)
\leq \big( \frac{1}{\pi }\big) ^p|t-s|^pK_{s}^p\| f\| _{p,\alpha }^p
\end{equation*}
and so
\begin{equation*}
\| Q_{t}(f)-Q_{s}(f)\| _{p,\alpha }\leq \frac{1}{\pi }|t-s|K_{s}\|
f\| _{p,\alpha }.
\end{equation*}
Then $(Q_{t},t\geq 0)$ is a ($C_{0}$)-contraction semigroup of linear
operators on $B_{\alpha }^p(D)$. Also, since the series representation for
$Q_{t}(f)(z)$ is uniformly convergent in any compact disk included in $D$,
it can be differentiated term by term, with respect to $t$ and $\varphi $.
We then easily obtain that $Q_{t}(f)(z)$ satisfies the Cauchy problem in the
statement of the theorem. We also note that in equation we must take $
z\not=0 $ simply because $z=0$ has no polar representation. This completes
the proof.
\end{proof}

Reasoning exactly as in the proof of
 \cite[Theorem 2.3.1, (v), pp. 53-54]{GGG3}, we immediately get the following
result.

\begin{corollary}\label{cor3}
Let $1\leq p<+\infty $. For each $f\in B_{\alpha}^p(D) $, the initial
value problem
\begin{equation*}
\frac{\partial u}{\partial t}+z\frac{\partial u}{\partial z}=0,\quad
(t,z)\in \mathbb{R}_{+}\times D\backslash \{0\},\quad u(0,z)=f(z),\;z\in D,
\end{equation*}
is well-posed and its unique solution is 
$Q_{t}(f)\in C^{\infty }(\mathbb{R}_{+};B_{\alpha }^p(D))$.
\end{corollary}

\begin{remark} \label{rmk3.3} \rm
The above results can be easily extended to several complex variables. We
may define the complex Poisson-Cauchy integral by
\begin{align*}
P_{t}(f)(z_{1},\dots ,z_{n})
& =\frac{\Gamma ((n+1)/2)}{\pi ^{(n+1)/2}}
t\int_{-\infty }^{+\infty }\dots \int_{-\infty }^{+\infty
}f(z_{1}e^{-iu_{1}},\dots ,z_{n}e^{-iu_{n}}) \\
&\quad  \times \frac{1}{(t^{2}+u_{1}^{2}+\dots +u_{n}^{2})^{(n+1)/2}}du_{1}
\dots du_{n}.
\end{align*}
Using similar arguments, as in the univariate complex case, we can prove
that the unique solution of the Cauchy problem
\begin{gather*}
\frac{\partial ^{2}u}{\partial t^{2}}(t,z_{1},\dots ,z_{n})+\frac{{\partial }
^{2}u}{\partial {\varphi _{1}}^{2}}(t,z_{1},\dots ,z_{n})+\dots +\frac{{\partial }
^{2}u}{\partial {\varphi _{n}}^{2}}(t,z_{1},\dots ,z_{n})=0,\quad t>0, \\
z_{1}=r_{1}e^{i\varphi _{1}},\dots ,\text{ }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}),\quad \text{for }
 z_{1},\dots ,z_{n}\in D,\; f\in B_{\alpha }^p(D^{n})
\end{gather*}
is given by
$u(t,z_{1},\dots ,z_{n})=P_{t}(f)(z_{1},\dots ,z_{n})$.
\end{remark}

\begin{remark} \label{rmk3.4} \rm
Reasoning as in the proof of Corollary \ref{cor3} and calculating
\begin{equation*}
\Big( \frac{d}{dt}P_{t}(f)(z_{1},\dots ,z_{n})\Big)\Big| _{t=0},
\end{equation*}
we easily obtain the following result:
For each $f\in B_{\alpha}^p(D^{n})$, the initial value problem
\begin{gather*}
\frac{\partial u}{\partial t}+\sum_{k=1}^{n}\big( z_{k}\frac{
\partial u}{\partial z_{k}}\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 $Q_{t}(f)\in C^{\infty }(\mathbb{R}
_{+};B_{\alpha }^p(D))$.
\end{remark}

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\end{document}