Two weight inequality for Hankel form on weighted Bergman spaces induced by doubling weights

The boundedness of the small Hankel operator $h_f^\nu(g)=P_\nu(f\bar{g})$, induced by an analytic symbol $f$ and the Bergman projection $P_\nu$ associated to $\nu$, acting from the weighted Bergman space $A^p_\om$ to $A^q_\nu$ is characterized on the full range $0<p,q<\infty$ when $\omega,\nu$ belong to the class $\mathcal{D}$ of radial weights admitting certain two-sided doubling conditions. Certain results obtained are equivalent to the boundedness of bilinear Hankel forms, which are in turn used to establish the weak factorization $A_{\eta}^{q}=A_{\omega}^{p_{1}}\odot A_{\nu}^{p_{2}}$, where $1<q,p_{1},p_{2}<\infty$ such that $q^{-1}=p_{1}^{-1}+p_{2}^{-1}$ and $\widetilde{\eta}^{\frac{1}{q}}\asymp\widetilde{\omega}^{\frac{1}{p_{1}}}\widetilde{\nu}^{\frac{1}{p_{2}}}$. Here $\widetilde{\tau}(r)=\int_r^1\tau(t)\,dt/(1-t)$ for all $0\le r<1$.