Zero sequences, factorization and sampling measures for weighted Bergman spaces

The zero sets of the Bergman space $A_{\omega }^p$ induced by either a radial weight $\omega$ admitting a certain doubling property or a non-radial Bekoll'e-Bonami type weight are characterized in the spirit of Luecking’s results from 1996. Accurate results obtained en route to this characterization are used to generalize Horowitz’s factorization result from 1977 for functions in $A_{\omega }^p$. The utility of the obtained factorization is illustrated by applications to integration and composition operators as well as to small Hankel operator induced by a conjugate analytic symbol. Dominating sets and sampling measures for the weighted Bergman space $A_{\omega }^p$ induced by a doubling weight are also studied. Several open problems related to the scheme of the paper are posed.