  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 $A^p_\omega$ 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 $\int_{r}^1\omega(s)\,ds\le
C\int_{\frac{1+r}{2}}^1\omega(s)\,ds$.
