Hankel forms and measures on weighted Bergman spaces

TL;DR

This paper characterizes the boundedness of Hankel forms and operators on weighted Bergman spaces with upper-doubling weights, providing new insights into measures and duality in these function spaces.

Contribution

It offers new characterizations of Hankel boundedness and measures on weighted Bergman spaces, extending existing theory and simplifying recent duality results.

Findings

Characterization of Hankel form boundedness on weighted Bergman spaces.

Identification of $A^p_ ext{omega}$ Hankel measures for $p \,\leq\, 2$.

Simplification of $A^1$ duality for small Bergman spaces.

Abstract

We characterize the boundedness of Hankel forms and Hankel operators induced by measures on weighted Bergman spaces, where the weights satisfy an upper-doubling condition. We also characterize $A^p_\omega$ Hankel measures for $p\leq 2$. The proofs leverage the existing theory of weighted Bergman spaces and the recent results on two-weight fractional derivatives, also simplifying the recent $A^1$ duality for small Bergman spaces obtained by Pel\'aez and R\"atty\"a.