For \(0<p,q<\infty \) and \(\omega \) a radial weight, the space \(L^{p,q}_\omega \) consists of those complex-valued measurable functions f on the unit disk such that \(\begin{aligned} \Vert f\Vert _{L^{p,q}_\omega }^q = \int _0^1 \left( \frac{1}{2\pi }\int _0^{2\pi }|f(re^{i\theta })|^pd\theta \right) ^{\frac{q}{p}}r\omega (r)\,dr, \end{aligned}\) and the mixed norm space \(A^{p,q}_\omega \) is the subset of \(L^{p,q}_\omega \) consisting of analytic functions. We say that a radial weight \(\omega \) belongs to \(\widehat{{\mathcal {D}}}\) if there exists \(C=C(\omega )>0\) such that \(\begin{aligned} \int _r^1\omega (s)ds \le C \int _{\frac{1+r}{2}}^1\omega (s)\,ds, \quad 0\le r <1. \end{aligned}\) We describe the dual space of \(A^{p,q}_\omega \) for \(0<p,q<\infty \) and \(\omega \in \widehat{{\mathcal {D}}}\) . Later on, we apply the obtained description of the dual space of \(A^{p,q}_\omega \) to prove that the Bergman projection induced by \(\omega \) , \(P_\omega \) , is bounded on \(L^{p,q}_\omega \) for \(1<p,q<\infty \) and \(\omega \in \widehat{{\mathcal {D}}}\) . Besides, if \(\omega \in \widehat{{\mathcal {D}}}\) and \(1<p,q<\infty \) , we also prove that \(P_\omega \) and the corresponding maximal Bergman projection \(P_\omega ^+\) are simultaneously bounded on \(L^{p,q}_\omega \) if and only if \(\omega \in {\mathcal {D}}.\)
