Essential normality of quotient modules vs. Hilbert-Schmidtness of submodules in H²( D²)

In the present paper, we prove that all the quotient modules in H²( D²), 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 H²( D²). Moreover, we obtain the equivalence of the essential normality of a quotient module and the Hilbert-Schmidtness of its associated submodule in H²( D²), 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 {Ya3}.