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 $\mathrm{\Psi }$ to provide $W_{\mathrm{\Psi },\phi }$ complex symmetric, employing a unique conjugation $\mathcal{J}$.