On the isomorphism classes of weighted spaces of harmonic and holomorphic functions

Let Ω be either the complex plane or the open unit disc. We completely determine the isomorphism classes of Hv = { f : Ω → C holomorphic : sup z∈Ω |f(z)|v(z) < ∞ } and investigate some isomorphism classes of hv = { f : Ω → C harmonic : sup z∈Ω |f(z)|v(z) < ∞ } where v is a given radial weight function. Our main results show that, without any further condition on v, there are only two possibilities for Hv, namely either Hv ∼ l∞ or Hv ∼ H∞, and at least two possibilities for hv, again hv ∼ l∞ and hv ∼ H∞. We also discuss many new examples of weights.