Complex symmetric weighted composition operators on the space $\mathcal{H}^2_2(\mathbb{D})$

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 $\mathcal{J}$.