Polynomial density in Bers spaces

Abstract

Let D be a bounded Jordan domain such that ∫ ∫ D λ D 2 − q d x d y > ∞ \smallint \;\smallint {\;_D}\lambda _D^{2 - q}\;dx\;dy\; > \infty for q > 1 q > 1 . Here λ D ( z ) {\lambda _D}(z) is the Poincaré metric for D. Define A q p ( D ) A_q^p(D) , the Bers space, to be the Fréchet space of holomorphic functions f on D, such that ‖ f ‖ q , p p = ∫ ∫ D λ D 2 − q p | f | p d x d y \left \| f \right \|_{q,p}^p = \smallint \;\smallint {\;_D}\lambda _D^{2 - qp}|f{|^p}\;dx\;dy is finite, 0 > p > ∞ , q p > 1 0 > p > \infty ,qp > 1 . It is well known that the polynomials are dense in A q p ( D ) A_q^p(D) for q p ⩾ 2 qp \geqslant 2 . We show that they are dense in A q p ( D ) A_q^p(D) for q p > 1 qp > 1 irrespective whether the boundary of D is rectifiable or not.