Polydisc version of Arveson's conjecture

TL;DR

This paper proves the polydisc version of Arveson's conjecture by establishing criteria for essential normality of homogeneous quotient modules in Hardy and weighted Bergman spaces over the polydisc.

Contribution

It provides a complete characterization of essential normality for homogeneous quotient modules on the polydisc, extending to weighted Bergman modules.

Findings

Criteria for essential normality of quotient modules established

Method applies to Hardy and weighted Bergman modules

Advances understanding of Arveson's conjecture in multivariable settings

Abstract

In the present paper, we solve the polydisc-version of Arveson Conjecture by giving a complete criteria for essential normality of homogeneous quotient modules of the Hardy module over the polydisc, and it turns out that our method applies to quotient modules of the weighted Bergman modules $A^2_s(\mathbb D^d)$.