  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.
