Weighted L^p → L^q-boundedness of commutators and paraproducts in the Bloom setting

Abstract

As our main result, we supply the missing characterization of the L^p(μ)→L^q(λ) boundedness of the commutator of a non-degenerate Calderón–Zygmund operator T and pointwise multiplication by b for exponents 1<q<p<∞ and Muckenhoupt weights μ∈A_p and λ∈A_q. Namely, the commutator [b,T]:L^p(μ)→L^q(λ) is bounded if and only if b satisfies the following new, cancellative condition: M_ν^#b∈L^pq/(p−q)(ν), where M_ν^#b is the weighted sharp maximal function defined by [Formula prsented] and ν is the Bloom weight defined by ν^1/p+1/q′:=μ^1/pλ^−1/q. In the unweighted case μ=λ=1, by a result of Hytönen the boundedness of the commutator [b,T] is, after factoring out constants, characterized by the boundedness of pointwise multiplication by b, which amounts to the non-cancellative condition b∈L^pq/(p−q). We provide a counterexample showing that this characterization breaks down in the weighted case μ∈A_p and λ∈A_q. Therefore, the introduction of our new, cancellative condition is necessary. In parallel to commutators, we also characterize the weighted boundedness of dyadic paraproducts Π_b in the missing exponent range p≠q. Combined with previous results in the complementary exponent ranges, our results complete the characterization of the weighted boundedness of both commutators and of paraproducts for all exponents p,q∈(1,∞).