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 $A_{\text{om}}^p$ to $A_{\nu }^q$ is characterized on the full range $0<p,q<\mathrm{\infty }$ 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 $1<q,p_1,p_2<\mathrm{\infty }$ such that $q^{-1}=p_1^{-1}+p_2^{-1}$ and ${\tilde{\eta }}^{\frac{1}{q}}\asymp {\tilde{\omega }}^{\frac{1}{p_1}}{\tilde{\nu }}^{\frac{1}{p_2}}$. Here $\tilde{\tau }(r)={\int }_r^1\tau (t)dt\mathrm{/}(1-t)$ for all $0\le r<1$.