Duality of mixed norm spaces induced by radial one-sided doubling weight

For 0<p,q<∞ and $ω$ a radial weight, the space Lp,q_ω consists of complex-valued measurable functions f on the unit disk such that \| f\|_Lp,q_ωq = ∫₀¹ (1/2π∫₀2π|f(reiθ)|ᵖdθ )q/prω(r)\,dr, and the mixed norm space Ap,q_ω is the subset of Lp,q_ω consisting of analytic functions. We say that a radial weight $ω$ belongs to D if there exists $C=C(ω)>0$ such that ∫ᵣ¹ω(s)ds ≤ C ∫1+r/2¹ω(s)\,ds \,\, for every\,\, 0≤ r <1. We describe the dual space of Ap,q_ω for every 0<p,q<∞ and ω∈D. Later on, we apply the obtained description of the dual space of Ap,q_ω to prove that the Bergman projection induced by $ω$, P_ω, is bounded on Lp,q_ω for 1<p,q<∞ and ω∈ D. Besides, we also prove that P_ω and the corresponding maximal Bergman projection P_ω⁺ are not simultaneously bounded on Lp,q_ω for 1<p,q<∞ and ω∈ D.