on the essential normality of principal submodules of the Drury-Arveson module

Continuing our earlier investigation [15] of the essential normality of submodules generated by polynomials, the emphasis of this paper is on submodules of the Drury-Arveson module H _n ² . In the case of two complex variables, we show that for every polynomial q ∈ C[z ₁ , z ₂ ], the submodule [q] of H ₂ ² is p-essentially normal for p > 2. In the case of three complex variables, we show there is a significant class of q ∈ C[z ₁ , z ₂ , z ₃ ] for which the submodule [q] of H ₃ ² is p-essentially normal for p > 3. The difficulties involved in the proofs of these results are determined by the weight t (−n ≤ t < ∞) of the space involved. Our earlier paper [15] covered the range −2 < t < ∞, which was enough to settle the problem for all polynomial-generated submodules of the Hardy module H ² (S). In this paper, we first solve the problem unconditionally for the weight range −3 < t ≤ −2, a consequence of which is the H ₂ ² -result mentioned above. We then consider the weight t = −3, which requires a substantial amount of additional work. At the moment we are only able to solve the t = −3 problem under a technical restriction on q, giving us the partial H ₃ ² -result mentioned above.