Essential normality of quotient modules vs. Hilbert-Schmidtness of submodules in $H^2({\mathbb{D}}^2)$

In the present paper, we prove that all the quotient modules in $H^2({\mathbb{D}}^2)$, associated to the finitely generated submodules containing a distinguished homogenous polynomial, are essentially normal, which is the first result on the essential normality of non-algebraic quotient modules in $H^2({\mathbb{D}}^2)$. Moreover, we obtain the equivalence of the essential normality of a quotient module and the Hilbert-Schmidtness of its associated submodule in $H^2({\mathbb{D}}^2)$, in the case that the submodule contains a distinguished homogenous polynomial. As an application, we prove that each finitely generated submodule containing a polynomial is Hilbert-Schmidt, which partially gives an affirmative answer to the conjecture of Yang \cite{Ya3}.