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

TL;DR

This paper characterizes the boundedness of small Hankel operators between weighted Bergman spaces with doubling weights, establishing weak factorization results and connections to bilinear forms for a broad parameter range.

Contribution

It provides a full characterization of Hankel operator boundedness on weighted Bergman spaces with doubling weights and links these to weak factorization and bilinear form boundedness.

Findings

Characterization of Hankel operator boundedness for all p,q in (0,∞)

Establishment of weak factorization for weighted Bergman spaces

Equivalence of boundedness to bilinear Hankel form boundedness

Abstract

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 $0\le r<1$.