We characterize the boundedness of the commutators
with biparameter
Journé operators
in the two-weight, Bloom-type setting, and express the norms of these
commutators in terms of a weighted little bmo norm of the symbol
. Specifically,
if
and
are biparameter
weights,
is the Bloom
weight, and
is in
,
then we prove a lower bound and testing condition
,
where
and
are Riesz transforms acting in each variable. Further, we prove that for such symbols
and any biparameter
Journé operators
,
the commutator
is bounded. Previous results in the Bloom setting do not include the biparameter
case and are restricted to Calderón–Zygmund operators. Even in the unweighted,
case,
the upper bound fills a gap that remained open in the multiparameter literature for
iterated commutators with Journé operators. As a by-product we also obtain a
much simplified proof for a one-weight bound for Journé operators originally due to
R. Fefferman.
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