Littlewood-Paley estimates for weighted Bergman spaces

In this talk, we consider the following Littlewood-Paley estimates for weighted Bergman spaces: for $p>0$, characterize positive weight functions $\omega(z)$ on the unit disk $\mathbb{D}$ such that $$\int_\mathbb{D} |f(z)|^p \omega(z) dA(z)<\infty$$ if and only if $$\int_\mathbb{D} |f'(z)|^p (1-|z|^2)^p \omega(z) dA(z)<\infty,$$ where $f$ is analytic in $\mathbb{D}$ and $dA$ is the area measure on $\mathbb{D}$. A complete solution to this problem is still lacking. We will talk about some results of these Littlewood-Paley estimates and give an application to double integral estimates for Besov type spaces.