We know that there are functions $f \in {A}(\mathbb{D}) = C(\overline{\mathbb{D}}) \cap \mathrm{Hol}(\mathbb{D})$ such that $f(z) = \sum_{k\geq0}a_k z^k$ on $\mathbb{D}$ with $a=(a_k)_{k\geq0} \notin \ell^1(\mathbb{N})$. This happens because $\imath \colon \ell^1(\mathbb{N}) \to {A}(\mathbb{D})$ given by $\imath(a) = \sum_{k\geq0} a_kz^k$ cannot be surjective.
Are there any characterisations of $\imath(\ell^1(\mathbb{N}))$? In other words, I have $g \in {A}(\mathbb{D})$; when does the power series of $g$ have summable coefficients?
Also, if $g$ is holomorphic on $\mathbb{D}^m$, $m\geq 2$, and continuous on its closure, when does $g(z_1,\dots,z_m) = \sum_{h_1,\dots,h_m\geq0} a_{h_1\cdots h_m}z_1^{h_1}\cdots z_m^{h_m}$ with $a \in \ell^1(\mathbb{N}^m)$?