  In this paper, we introduce a new norm for $\mathcal{S}^2(\mathbb{D})$,
encompassing functions whose first and second derivatives belong to both the
Hardy space $\mathcal{H}^2(\mathbb{D})$ and the classical Bergman space
$\mathcal{A}^2(\mathbb{D})$. Moreover, we present some basic properties of the
space $\mathcal{H}^2_2(\mathbb{D})$ and subsequently establish conditions for
symbols $\phi$ and $\Psi$ to provide $W_{\Psi, \phi}$ complex symmetric,
employing a unique conjugation $\mathcal{J}$.
