Paper detail

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

The boundedness of the small Hankel operator $h_f^ν(g)=P_ν(f\bar{g})$, induced by an analytic symbol $f$ and the Bergman projection $P_ν$ associated to $ν$, acting from the weighted Bergman space $A^p_\om$ to $A^q_ν$ is characterized on the full range $0<p,q<\infty$ when $ω,ν$ 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 $A_η^{q}=A_ω^{p_{1}}\odot A_ν^{p_{2}}$, where $1<q,p_{1},p_{2}<\infty$ such that $q^{-1}=p_{1}^{-1}+p_{2}^{-1}$ and $\widetildeη^{\frac{1}{q}}\asymp\widetildeω^{\frac{1}{p_{1}}}\widetildeν^{\frac{1}{p_{2}}}$. Here $\widetildeτ(r)=\int_r^1τ(t)\,dt/(1-t)$ for all $0\le r<1$.