On polynomial density in 𝐴_{𝑞}(𝐷)

Abstract

Let D D be a bounded Jordan domain. Define A q ( D ) {A_q}(D) , the Bers space, to be the Banach space of holomorphic functions on D D , such that ∬ D | f | λ D 2 − q d x d y \iint _D {|f|\lambda _D^{2 - q}dxdy} is finite, where λ D ( z ) {\lambda _D}(z) is the Poincaré metric for D D . It is well known that the polynomials are dense in A q ( D ) {A_q}(D) for 2 ≦ q > ∞ 2 \leqq q > \infty and we shall prove they are dense in A q ( D ) {A_q}(D) for 1 > q > 2 1 > q > 2 if the boundary of D D is rectifiable. Also some remarks are made in case the boundary of D D is not rectifiable.