Bergman space $L_a^2(\mathbb C)$

Question

I claim that the Bergman space $L_a^2(\mathbb C)$ is the zero space. Is this true? If it is, how can I prove that every non-constant entire function is not in $L^2$?

Answer 1

Here is an answer, assuming that you are considering the unweighted $L^2$ norm. If you tell me what $a$ is, I can make an edit addressing that case.

Without loss of generality, assume $f(0) = 1$ (we can shift and scale $f$ if not).

Then

$\begin{align*} 1 &= |f(0)|\\ &=\left|\frac{1}{\pi r^2} \int_{D(r)} f(z) dA\right|\\ &\leq \frac{1}{\sqrt{\pi r^2}} \left[ \int_{D(r)} |f(z)|^2 dA\right]^{\frac{1}{2}} \end{align*}$

So we see for any positive constant $r$, $\lVert f \rVert_{L^2(D(r))} \geq \sqrt{\pi} r$, which then implies that $f$ is not in $L^2(\mathbb{C})$.

Let me know if any part of this is unclear to you.