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 is an element of 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 is an element of C[Z(1), Z(2), Z(3)] for which the submodule [q] of is H-3(2) p-essentially normal for p > 3. The difficulties involved in the proofs of these results are determined by the weight t (-n <= t < infinity) of the space involved. Our earlier paper [15] covered the range -2 < t < infinity, 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.