Borel transforms of functions in parametrized family of Hilbert spaces

Abstract

We consider Hilbert spaces of entire functions
\begin{equation*}
P_\beta (D)=\left \{F\in H(\mathbb{C}):\ \|F\|^2:=\int\limits_0^{2\pi }\int\limits_0^\infty \frac {|F(re^{i\varphi })|^2drd\Delta (\varphi)}{K(re^{i\varphi })r^{2\beta }}<\infty \right \},
\end{equation*}
where $D$ is a bounded convex domain on the complex plane,
\begin{align*}
&K(\lambda)=\|e^{\lambda z}\|^2_{L_2(D)}=\int\limits_D|e^{\lambda z}|^2dm(z),\quad \lambda \in \mathbb{C},
\\
&h(\varphi)=\max_{z\in \overline D} \hbox{Re}\, ze^{i\varphi },\quad \varphi \in [0;2\pi ],
\\
&\Delta (\varphi)=h(\varphi)+\int\limits_{0}^\varphi h(\theta)d\theta,\quad \varphi \in [0;2\pi ].
\end{align*}
The interest to these spaces is motivated by the fact that $P_0(D)$ is the space of Laplace transforms of linear continuous functionals on the Bergman space $B_2(D)$, while $P_{\frac 12}(D)$ is the space of Laplace transforms of linear continuous functionals on the Smirnov space $E_2(D)$. In the paper for the parameters $\beta \in \left (-\frac 12;\frac 32\right)$ we provide a complete description of the Borel transforms of functions in spaces $P_\beta (D)$. In this way, the Bergman and Smirnov spaces are
embedded into a scale of Hilbert spaces and, in the authors' opinion, this could allow
to apply the theory of Hilbert scales for studying the problems in these spaces.