Let $S=\{z\in \mathbb{C}: |z|=1\}$ the circle of $\mathbb{C}$. We consider the applications $h^{p}(z)$ defined on $C\ni z\to\mathbb{C}$ with $(p\in \mathbb{N})$ such that:
$h^{p}(rz)=r^ph^{p}(z),\quad r>0$ and $\int_{S}h^{p}(z)d\sigma(z)=0$ where $d\sigma(w)$ is the measure Lebesgue sur on $S$.
$\int_{S}|h^{p}(z)|^2d\sigma(z)=1$ and $\int_{S} h^{p}(z)\overline{h^{q}(z)}d\sigma(z)=\delta_{p,q}$.
They $h^{p}$ are called spherical harmonic.
Now we define $P(z,w)=\sum_{p=0}^{\infty} \frac{1}{\sqrt{p}!}h^{p}(z)\overline{h^{p}(w)}\exp(-\frac{|z|^2}{2})$ with $z,w \in C$, and the transformation $$P(g)(z)=\int_{S^1}g(w)P(z,w)d\sigma(w)$$.
Put $\Gamma:=\{ g(w)=\sum_{p\geq0} a^ph^p(w) (w\in S): ||g||^2=\int_{S} |g(w)|^2d\sigma(w)=\sum^{\infty}_{p\geq0}|a^p|^2<\infty \}$ where $a_p\in \mathbb{C}$
For $g\in \Gamma$ i.e., $g(w)=\sum_{p\geq0} a^ph^p(w), w\in S$ a simple computation shows : $||P(g)||^2_{L^2(\mathbb{C})}=\int_{\mathbb{C}}|P(g)(z)|^2dz=\frac{1}{2}||g||^2<\infty$ where $dz$ is the measure de Lebesgue on $\mathbb{C}$. Suppose that $\Gamma$ is an isomorph to $(\Gamma, ||_S)$ onto à $(A,L^2(\mathbb{C}))$ where $A:=P(\Gamma)$
Now, if we put $\Gamma_r=\{g\in \Gamma: \int_{S^1}|g(w)|^rd\sigma(w) <\infty\},\quad r>2$ then $\Gamma_r\subset \Gamma$ .
Is there a characterization of $\Gamma_r$ ? as $\Gamma$ and can we find a function $\psi(z)$ such that $$P(g)(z)=\int_{S^1}|P(z,w)|d\sigma(w)<\psi(z)$$
Any indications will be appreciated