Let $f$ be a bounded holomorphic function on $\mathbb D^2$ and $s : \mathbb C^2 \longrightarrow \mathbb C^2$ be the symmetrization map given by $s(z) = (z_1 + z_2, z_1 z_2),$ for $z = (z_1, z_2) \in \mathbb C^2.$ If $\{(f \circ s) g_n\}_{n \geq 1}$ has a convergent subsequence in $L^2 (\mathbb D^2)$ for any sequence of bounded anti-symmetric holomorphic square integrable functions $\{g_n\}_{n \geq 1}$ then what can we conclude when $g_n$'s are replaced by a sequence of bounded symmetric square integrable holomorphic functions?
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