On the boundedness of Bergman projection

The main purpose of this survey is to gather results on the boundedness of the Bergman projection. First, we shall go over some equivalent norms on weighted Bergman spaces $A_{\omega }^p$ which are useful in the study of this question. In particular, we shall focus on a decomposition norm theorem for radial weights~$\omega$ with the doubling property $\int_{r}^1\omega(s),ds\le C\int_{\frac{1+r}{2}}^1\omega(s),ds$.