The hypotheses
I have the following series : $ f(z)=f(rw)=\sum^{+\infty}_{k=0}r^{k}f_{k}(r)h_{k}(w)$ with $z\in\Bbb{C}^{2}$ which converges uniformly on any compact of $\Bbb{C}^{2}$ where
1) $z=rw$ is the polar coordinate writing of $z$, with $r>0$, $w\in S$ and $S=\{ (z,z')\in \Bbb{C}^{2}\ : |z|^{2}+|z'|^{2}=1\}$ the sphere of $\Bbb{C}^{2}$.
2) For $k\in\Bbb{N}$, $f_{k}$ is an application of $\Bbb{R}$ to $\Bbb{R}$.
3) $\int^{+\infty}_{0} r^{3} \big|\quad f_{0}(r)\quad \big|^{p}dr=+\infty$ for $p\geqslant 1$.
4)For $k\in\Bbb{N}$, $h_{k}$ is an application of $S$ to $\Bbb{C}$ such that:
$$ \int_{S}h_{k}(w)d\sigma(w)=0\quad if \quad k>0 \quad, and \quad \int_{S}h_{0}(w)d\sigma(w)\not =0$$
$d\sigma(w)$ is Lebesgue's measure on the sphere $S$.
I want to show that $f$ is not in $L^{p}(\Bbb{C}^{2},dz)\quad p\geqslant 1$ with $dz$ is the measure of Lebesgue on $\Bbb{C}^{2}$. To do this I reason with the absurd, i.e I suppose
$$ \|f\|^{p}=\int_{\Bbb{C}^{2}}|f(z)|^{p}dz<+\infty $$
Then, by passing in polar coordinates we obtain:
$$\|f\|^{p}=\int^{+\infty}_{0}r^{3}\int_{S}|f(rw)|^{p}d\sigma(w)dr<+\infty.$$
So, by Fubini's Theorem, we have : For almost $r>0$, we have: $r^{3}\int_{S}|f(rw)|^{p}d\sigma(w)<+\infty$ and for almost $w$, we have: $\int^{+\infty}_{0}r^{3}|f(rw)|^{p}dr<+\infty$.
Now, by Hölder's inequality we have $$ const.|\int_{S}f(rw)d\sigma(w) |^{p}<\int_{S}|f(rw)|^{p}d\sigma(w) $$
and hence $$ const.\big|\int_{S} \quad \sum^{+\infty}_{k=0}r^{k}f_{k}(r)h_{k}(w) \quad d\sigma(w)\quad\big|^{p}<\int_{S}|f(rw)|^{p}d\sigma(w)$$
Since the convergence is uniform in $w$, we can change the integral and the sum, so $$const.\big|\quad \sum^{+\infty}_{k=0}r^{k}f_{k}(r)\int_{S} h_{k}(w) \quad d\sigma(w) \quad \big|^{p}<\int_{S}|f(rw)|^{p}d\sigma(w).$$
By using the hypothesis $4$, we have $$const.\big|\quad f_{0}(r)\quad \big|^{p}<\int_{S}|f(rw)|^{p}d\sigma(w)$$
Hence $$const r^{3}\big|\quad f_{0}(r)\quad \big|^{p}<r^{3}\int_{S}|f(rw)|^{p} d\sigma(w)$$
Finally, since $\int^{+\infty}_{0} r^{3}\big|\quad f_{0}(r)\quad \big|^{p}dr=\infty$ we get $\int^{+\infty}_{0}r^{3}\int_{S}|f(rw)|^{p} d\sigma(w)dr=\infty$ absurd because by Fubini's theorem we have $$\int^{+\infty}_{0}r^{3}|f(rw)|^{p}dr<+\infty.$$
Thanks for the reading