  Let $v(r)=\exp\left(-\frac{\alpha}{1-r}\right)$ with $\alpha>0$, and let
$\mathbb{D}$ be the unit disc in the complex plane. Denote by $A^p_v$ the
subspace of analytic functions of $L^p(\mathbb{D},v)$ and let $P_v$ be the
orthogonal projection from $L^2(\mathbb{D},v)$ onto $A^2_v$. In 2004, Dostanic
revealed the intriguing fact that $P_v$ is bounded from $L^p(\mathbb{D},v)$ to
$A^p_v$ only for $p=2$, and he posed the related problem of identifying the
duals of $A^p_v$ for $p\ge 1$, $p\neq 2$. In this paper we propose a solution
to this problem by proving that $P_v$ is bounded from $\,L^p(\D,v^{p/2})$ to
$A^p_{v^{p/2}}$ whenever $1\le p <\infty$, and, consequently, the dual of
$A^p_{v^{p/2}}$ for $p\ge 1$ can be identified with $A^{q}_{v^{q/2}}$, where
$1/p+1/q=1$. In addition, we also address a similar question on some classes of
weighted Fock spaces.
