Bergman projection induced by radial weight acting on growth spaces

Let ω be a radial weight on the unit disc of the complex plane D and denote ωₓ =∫₀¹ sˣ ω(s)\,ds, x≥ 0, for the moments of ω and ω(r)=∫ᵣ¹ ω(s)\,ds for the tail integrals. A radial weight ω belongs to the class D if satisfies the upper doubling condition 0<r<1ω(r)ω(1+r/2)<∞. If ν or ω belongs to D, it is described the boundedness of the Bergman projection P_ω induced by ω on the growth space L^∞_ν =\ f: \|f\|∞,v= esssupz |f(z)|ν(z)<∞\ in terms of neat conditions on the moments and/or the tail integrals of ω and ν. Moreover, it is solved the analogous problem for P_ω from L^∞_ν to the Bloch type space B^∞_ν of analytic functions such that z∈ D(1-|z|)ν(z) |f'(z)|<∞. We also study similar questions for exponentially decreasing radial weights.