Reproducing Kernel Thesis of Hankel Operators on Weighted Hardy Spaces

TL;DR

This paper extends the Reproducing Kernel Thesis for Hankel operators to the two-weight setting on weighted Hardy spaces, establishing new norm equivalences and boundedness criteria.

Contribution

It introduces a two-weight Garsia norm for Hankel operators and proves its equivalence to the weighted Garsia norm, generalizing the boundedness characterization.

Findings

Two-weight Garsia norm is equivalent to the weighted Garsia norm.

Testing Hankel operators on reproducing kernels suffices for boundedness.

Proved a two-weight version of the Carleson embedding theorem.

Abstract

We study the boundedness of Hankel operators between two weighted spaces, with Muckenhoupt weights. In particular, we consider whether the Reproducing Kernel Thesis for Hankel operators generalizes to the case of two different weights. There, Hankel operators are bounded on the Hardy space if and only if they are bounded when tested on reproducing kernel functions. The supremum of testing the Hankel operator on this special class of functions is called the Garsia norm of the symbol of the Hankel operator, known to be equivalent to the BMO norm of the operator. We formulate a two-weight version of the Garsia norm and prove that testing the Hankel operator on the same reproducing kernel functions and measuring the norm in the two-weight setting is sufficient to prove that the operator is bounded. In the process,we prove that the Garsia norm is equivalent to the weighted Garsia norm, when two weights are the same, and we prove the two-weight version of the Carleson embedding theorem.