This paper is a comprehensive study of the Bergman projection induced by a radial weight, which presents some known results on the topic, but also an especially new one, generalizations and solutions for some open problems. A radial weight on the unit disc \(\mathbb D\) is the extension to \(\mathbb D\) of an integrable function \(\omega\) on \([0,1]\) by \(\omega(z)=\omega(|z|)\), \(z\in\mathbb D\). The space \(L^p_\omega\), the Hardy space \(\mathcal{H}(\mathbb D)\), the Bloch space \(\mathcal B\), and the Bergman space \(A^p_\omega=L^p_\omega\cap\mathcal{H}(\mathbb D)\) are considered. For a radial weight \(\omega\), the orthogonal Bergman projection \(P_\omega\) from \(L^2_\omega\) to \(A^2_\omega\) is \[P_\omega(f)(z) =\int_{\mathbb D}f(\zeta)\overline{B^\omega_z(\zeta)}dA(\zeta),\] where \(B^\omega_z\) is the reproducing kernel of \(A^2_\omega\) and \(dA\) is the normalized area measure on \(\mathbb D\). The first main result of this paper (Theorem 1) characterizes the class of radial weights such that \(P_\omega: L^\infty \rightarrow \mathcal B\) is bounded, and establishes the corresponding natural duality relations, under the hypothesis of positivity of the tail integral \(\widehat{\omega}=\int_{|z|}^1\omega(s)ds>0\) for all \(z\in\mathbb D\). A radial weight \(\omega\) belongs to the class \(\widehat{\mathcal D}\) if \(\widehat{\omega}\) satisfies the property \(\widehat{\omega}(r) \le C\widehat{\omega}(\frac{1+r}{2})\) for some constant \(C = C(\omega)\ge 1\) and for all \(0 \le r <1\). For other characterizations the classes \(\check{\mathcal D}\) and \(\mathcal D=\widehat{\mathcal D}\cap\check{\mathcal D}\) are introduced. It is said that \(\omega\in\check{\mathcal D}\) if there exist constants \(K=K(\omega)>1\) and \(C=C(\omega)>1\) such that \(\widehat{\omega}(r)\ge C\widehat{\omega}(1-\frac{1-r}{K})\) for all \(0\le r<1\).
The paper has the following structure. In the first section the main results are presented, contained in eleven main theorems and two important corollaries, establishing characterizations of the radial weights \(\omega\) on the unit disc such that \(P_\omega:L^\infty\rightarrow\mathcal B\) is bounded and/or acts surjectively from \(L^\infty\) to the Bloch space \(\mathcal B\), or the dual of the weighted Bergman space \(A^1_\omega\) is isomorphic to the Bloch space under the \(A^2_\omega\)-pairing. Also the problem posed by M. R. Dostanić [Proc. Edinb. Math. Soc., II. Ser. 47, No. 1, 111–117 (2004; Zbl 1077.47020)] on describing the weights \(\omega\) such that \(P_\omega\) is bounded on the Lebesgue space \(L^p_\omega\) is solved under a weak regularity hypothesis on \(\omega\). With respect to Littlewood-Paley estimates, the radial weights \(\omega\) are characterized such that the norm of any function in \(A^p_\omega\) is comparable to the norm in \(L^p_\omega\) of its derivative times the distance from the boundary, solving another well-known problem on the area. The following six sections are dedicated to the very laborious proofs of the previous presented theorems and results. In the last section of the paper two open problems closely related to the presented results are discussed, and two conjectures are posed. The first problem concerns Littlewood-Paley estimates and the second the boundedness of the Bergman projection \(P_\omega\) on \(L^p_\omega\).