Compact Hankel Operators on Generalized Bergman Spaces of the Polydisc

Let ${vartheta}$ be a measure on the polydisc ${mathbb{D}^n}$ which is the product of n regular Borel probability measures so that ${vartheta([r,1)^ntimesmathbb{T}^n) >0 }$ for all 0 < r < 1. The Bergman space ${A^2_{vartheta}}$ consists of all holomorphic functions that are square integrable with respect to ${vartheta}$ . In one dimension, it is well known that if f is continuous on the closed disc ${overline{mathbb{D}}}$ , then the Hankel operator H _f is compact on ${A^2_vartheta}$ . In this paper we show that for n ≥ 2 and f a continuous function on ${{overline{mathbb{D}}}^n}$ , H _f is compact on ${A^2_vartheta}$ if and only if there is a decomposition f = h + g, where h belongs to ${A^2_vartheta}$ and ${lim_{ztopartialmathbb{D}^n}g(z)=0}$ .