Boundedness of functions in the Hardy space $H^2(\mathbb H)$

Question

Let $F: \mathbb H \to \mathbb C$ be a function in the Hardy space $H^2(\mathbb H)$. In other words, we have $$\sup_{y>0} \int_{-\infty}^\infty |F(x+iy)|^2 \, dx < \infty.$$ Let $f(x) = \lim_{y \downarrow 0} F(x+iy)$ be the boundary function. Such limit exists a.e. in $x$, and we know that $f \in L^2(\mathbb R, \mathbb C)$.

What I want is to make $F$ a bounded function on $\mathbb H$, by assuming additional condition if necessary. I think boundedness is not a very strong condition, considering the Phragmen-Lindelof principle.

For example, I would like to ask:

1. Is $F$ bounded?

2. If we further assume that $f\in L^\infty(\mathbb R, \mathbb C)$, then is $F$ bounded?

3. If we further assume that $f(x) \leq \frac{C}{(1+|x|)^\alpha}$ for some constants $C$ and $\alpha$, then is $F$ bounded?

Partial answers or other conditions are also welcome!