Weighted Lp → Lq-boundedness of commutators and paraproducts in the Bloom setting

As our main result, we supply the missing characterization of the $L^p\left(\mu \right)\to L^q\left(\lambda \right)$ boundedness of the commutator of a non-degenerate Calderón–Zygmund operator T and pointwise multiplication by b for exponents $1<q<p<\infty$ and Muckenhoupt weights $\mu \in A_p$ and $\lambda \in A_q$. Namely, the commutator $[b,T]:L^p\left(\mu \right)\to L^q\left(\lambda \right)$ is bounded if and only if b satisfies the following new, cancellative condition:$M_{\nu }^{\#}b\in L^{pq/(p-q)}\left(\nu \right),$ where $M_{\nu }^{\#}b$ is the weighted sharp maximal function defined by$M_{\nu }^{\#}b:=\underset{Q}{\sup }\frac{1_Q}{\nu \left(Q\right)}\underset{Q}{\int }|b-{〈b〉}_Q|dx$ and ν is the Bloom weight defined by ${\nu }^{1/p+1/q^{\prime }}:={\mu }^{1/p}{\lambda }^{-1/q}$.

In the unweighted case $\mu =\lambda =1$, 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 $b\in L^{pq/(p-q)}$. We provide a counterexample showing that this characterization breaks down in the weighted case $\mu \in A_p$ and $\lambda \in A_q$. Therefore, the introduction of our new, cancellative condition is necessary.

In parallel to commutators, we also characterize the weighted boundedness of dyadic paraproducts ${\Pi }_b$ in the missing exponent range $p\ne q$. 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 $p,q\in (1,\infty )$.