Weighted $L^p\to L^q$-boundedness of commutators and paraproducts in the Bloom setting

Abstract

As our main result, we supply the missing characterization of the $L^p(\mu)\to L^q(\lambda)$ boundedness of the commutator of a non-degenerate Calder\'on--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 $[b,T]\colon L^p(\mu)\to L^q(\lambda)$ is bounded if and only if $b$ satisfies the following new, cancellative condition: $$M^\#_\nu b\in L^{pq/(p-q)}(\nu),$$ where $M^\#_\nu b$ is the weighted sharp maximal function defined by $$ M^\#_\nu b:=\sup_{Q} \frac{\mathbf{1}_Q}{\nu(Q)} \int_{Q} |b-\langle b\rangle_Q |\,\mathrm{d}x$$ and $\nu$ is the Bloom weight defined by $\nu^{1/p+1/q'}:= \mu^{1/p} \lambda^{-1/q}$. In the unweighted case $\mu=\lambda=1$, by a result of Hyt\"onen 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 $p\neq q$. Combined with previous results in the complementary exponent ranges, our results complete the characterisation of the weighted boundedness of both commutators and of paraproducts for all exponents $p,q \in (1,\infty)$.