p ‐essential normality of quasi‐homogeneous Drury–Arveson submodules

The first author and Wang [ Math. Ann. 340 (2008) 907–934] proved that each homogeneous principal submodule of the Drury–Arveson moduleH n 2is essentially normal, and hence in dimensions n = 2, 3 each homogeneous submodule ofH n 2is essentially normal. For the Bergman modulesL a 2 ( B n )on the unit ball, Douglas and Wang [ J. Funct. Anal. 261 (2011) 3155–3180] recently proved that every principal submodule is essentially normal. In this paper, we develop some new techniques to prove the essential normality of Drury–Arveson's quasi‐homogeneous principal submodules, by a combination with the approach of Douglas and Wang. As a consequence, we prove that each quasi‐homogeneous submodule ofH n 2is essentially normal for dimensions n = 2, 3, and determine the related K ‐homology.