On a family of weighted $\overline\partial$-integral representations in the unit disc

For weighted $L^p$-classess of $C^1$-functions in the unit disc with weight function of the type $|w|^{2\gamma}\cdot(1-|w|^{2\rho})^{\alpha}$, we obtain a family of weighted $\overline{\partial}$-integral representations of the type $f = P(f) - T(\overline{\partial} f)$.